Matrix Cookbook

The Matrix Cookbook

Kaare Brandt PetersenMichael Syskind Pedersen

Version: October 3, 2005

What is this? These pages are a collection of facts (identities, approxima-tions, inequalities, relations, ...) about matrices and matters relating to them.It is collected in this form for the convenience of anyone who wants a quickdesktop reference .

Disclaimer: The identities, approximations and relations presented here wereobviously not invented but collected, borrowed and copied from a large amountof sources. These sources include similar but shorter notes found on the internetand appendices in books - see the references for a full list.

Errors: Very likely there are errors, typos, and mistakes for which we apolo-gize and would be grateful to receive corrections at [email protected].

Its ongoing: The project of keeping a large repository of relations involvingmatrices is naturally ongoing and the version will be apparent from the date inthe header.

Suggestions: Your suggestion for additional content or elaboration of sometopics is most welcome at [email protected].

Acknowledgements: We would like to thank the following for discussions,proofreading, extensive corrections and suggestions: Esben Hoegh-Rasmussenand Vasile Sima.

Keywords: Matrix algebra, matrix relations, matrix identities, derivative ofdeterminant, derivative of inverse matrix, differentiate a matrix.

1

CONTENTS CONTENTS

Contents

1 Basics 51.1 Trace and Determinants . . . . . . . . . . . . . . . . . . . . . . . 51.2 The Special Case 2x2 . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Derivatives 72.1 Derivatives of a Determinant . . . . . . . . . . . . . . . . . . . . 72.2 Derivatives of an Inverse . . . . . . . . . . . . . . . . . . . . . . . 82.3 Derivatives of Matrices, Vectors and Scalar Forms . . . . . . . . 92.4 Derivatives of Traces . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Derivatives of Structured Matrices . . . . . . . . . . . . . . . . . 12

3 Inverses 153.1 Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Exact Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Implication on Inverses . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Pseudo Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Complex Matrices 194.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Decompositions 225.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 225.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . 225.3 Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . . 23

6 Statistics and Probability 246.1 Definition of Moments . . . . . . . . . . . . . . . . . . . . . . . . 246.2 Expectation of Linear Combinations . . . . . . . . . . . . . . . . 256.3 Weighted Scalar Variable . . . . . . . . . . . . . . . . . . . . . . 26

7 Gaussians 277.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 Mixture of Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Special Matrices 338.1 Units, Permutation and Shift . . . . . . . . . . . . . . . . . . . . 338.2 The Singleentry Matrix . . . . . . . . . . . . . . . . . . . . . . . 348.3 Symmetric and Antisymmetric . . . . . . . . . . . . . . . . . . . 368.4 Toeplitz Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.5 Positive Definite and Semi-definite Matrices . . . . . . . . . . . . 388.6 Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Petersen & Pedersen, The Matrix Cookbook, Version: October 3, 2005, Page 2

CONTENTS CONTENTS

9 Functions and Operators 419.1 Functions and Series . . . . . . . . . . . . . . . . . . . . . . . . . 419.2 Kronecker and Vec Operator . . . . . . . . . . . . . . . . . . . . 429.3 Solutions to Systems of Equations . . . . . . . . . . . . . . . . . 439.4 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.5 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.6 Integral Involving Dirac Delta Functions . . . . . . . . . . . . . . 469.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A One-dimensional Results 47A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 48

B Proofs and Details 50B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


CONTENTS CONTENTS

Notation and Nomenclature

A MatrixAij Matrix indexed for some purposeAi Matrix indexed for some purposeAij Matrix indexed for some purposeAn Matrix indexed for some purpose or

The n.th power of a square matrixA1 The inverse matrix of the matrix AA+ The pseudo inverse matrix of the matrix A (see Sec. 3.6)A1/2 The square root of a matrix (if unique), not elementwise(A)ij The (i, j).th entry of the matrix AAij The (i, j).th entry of the matrix A

[A]ij The ij-submatrix, i.e. A with i.th row and j.th column deleteda Vectorai Vector indexed for some purposeai The i.th element of the vector aa Scalar

1 BASICS

1 Basics

(AB)1 = B1A1

(ABC...)1 = ...C1B1A1

(AT )1 = (A1)T

(A + B)T = AT + BT

(AB)T = BT AT

(ABC...)T = ...CT BT AT

(AH)1 = (A1)H

(A + B)H = AH + BH

(AB)H = BHAH

(ABC...)H = ...CHBHAH

1.1 Trace and Determinants

Tr(A) =

iAiiTr(A) =

ii, i = eig(A)

Tr(A) = Tr(AT )Tr(AB) = Tr(BA)

Tr(A + B) = Tr(A) + Tr(B)Tr(ABC) = Tr(BCA) = Tr(CAB)

det(A) =

ii i = eig(A)det(AB) = det(A) det(B)det(A1) = 1/ det(A)

1.2 The Special Case 2x2

Consider the matrix A

A =[

A11 A12A21 A22

]

Determinant and trace

det(A) = A11A22 A12A21Tr(A) = A11 + A22

Eigenvalues2 Tr(A) + det(A) = 0


1.2 The Special Case 2x2 1 BASICS

1 =Tr(A) +

Tr(A)2 4 det(A)

22 =

Tr(A)

Tr(A)2 4 det(A)2

1 + 2 = Tr(A) 12 = det(A)

Eigenvectors

v1 [

A121 A11

]v2

[A12

2 A11

]

Inverse

A1 =1

det(A)

[A22 A12A21 A11

]


2 DERIVATIVES

2 Derivatives

This section is covering differentiation of a number of expressions with respect toa matrix X. Note that it is always assumed that X has no special structure, i.e.that the elements of X are independent (e.g. not symmetric, Toeplitz, positivedefinite). See section 2.5 for differentiation of structured matrices. The basicassumptions can be written in a formula as

XklXij

= iklj

that is for e.g. vector forms,[xy

]

i

=xiy

[x

y

]

i

=x

yi

[xy

]

ij

=xiyj

The following rules are general and very useful when deriving the differential ofan expression ([12]):

A = 0 (A is a constant) (1)(X) = X (2)

(X + Y) = X + Y (3)(Tr(X)) = Tr(X) (4)

(XY) = (X)Y + X(Y) (5)(X Y) = (X) Y + X (Y) (6)(XY) = (X)Y + X (Y) (7)

(X1) = X1(X)X1 (8)(det(X)) = det(X)Tr(X1X) (9)

(ln(det(X))) = Tr(X1X) (10)XT = (X)T (11)XH = (X)H (12)

2.1 Derivatives of a Determinant

2.1.1 General form

det(Y)x

= det(Y)Tr[Y1

Yx

]

2.1.2 Linear forms

det(X)X

= det(X)(X1)T

det(AXB)X

= det(AXB)(X1)T = det(AXB)(XT )1


AdministratorPencil

2.2 Derivatives of an Inverse 2 DERIVATIVES

2.1.3 Square forms

If X is square and invertible, then

det(XT AX)X

= 2 det(XT AX)XT

If X is not square but A is symmetric, then

det(XT AX)X

= 2 det(XT AX)AX(XT AX)1

If X is not square and A is not symmetric, then

det(XT AX)X

= det(XT AX)(AX(XT AX)1 + AT X(XT AT X)1) (13)

2.1.4 Other nonlinear forms

Some special cases are (See [8, 7])

ln det(XT X)|X

= 2(X+)T

ln det(XT X)X+

= 2XT

ln | det(X)|X

= (X1)T = (XT )1

det(Xk)X

= k det(Xk)XT

2.2 Derivatives of an Inverse

From [17] we have the basic identity

Y1

x= Y1 Y

xY1

from which it follows

(X1)klXij

= (X1)ki(X1)jlaT X1b

X= XT abT XT

det(X1)X

= det(X1)(X1)T

Tr(AX1B)X

= (X1BAX1)T


2.3 Derivatives of Matrices, Vectors and Scalar Forms 2 DERIVATIVES

2.3 Derivatives of Matrices, Vectors and Scalar Forms

2.3.1 First Order

xT ax

=aT xx

= a

aT XbX

= abT

aT XT bX

= baT

aT XaX

=aT XT a

X= aaT

XXij

= Jij

(XA)ijXmn

= im(A)nj = (JmnA)ij

(XT A)ijXmn

= in(A)mj = (JnmA)ij

2.3.2 Second Order

Xij

klmn

XklXmn = 2

kl

Xkl

bT XT XcX

= X(bcT + cbT )

(Bx + b)T C(Dx + d)x

= BT C(Dx + d) + DT CT (Bx + b)

(XT BX)klXij

= lj(XT B)ki + kj(BX)il

(XT BX)Xij

= XT BJij + JjiBX (Jij)kl = ikjl

See Sec 8.2 for useful properties of the Single-entry matrix Jij

xT Bxx

= (B + BT )x

bT XT DXcX

= DT XbcT + DXcbT

X(Xb + c)T D(Xb + c) = (D + DT )(Xb + c)bT


2.3 Derivatives of Matrices, Vectors and Scalar Forms 2 DERIVATIVES

Assume W is symmetric, then

s(xAs)T W(xAs) = 2AT W(xAs)

s(x s)T W(x s) = 2W(x s)

x(xAs)T W(xAs) = 2W(xAs)

A(xAs)T W(xAs) = 2W(xAs)sT

2.3.3 Higher order and non-linear

XaT Xnb =

n1r=0

(Xr)T abT (Xn1r)T (14)

XaT (Xn)T Xnb =

n1r=0

[Xn1rabT (Xn)T Xr

+(Xr)T XnabT (Xn1r)T]

(15)

See B.1.1 for a proof.Assume s and r are functions of x, i.e. s = s(x), r = r(x), and that A is aconstant, then

xsT As =

[sx

]T(A + AT )s

xsT Ar =

[sx

]TAs +

[rx

]TAT r

2.3.4 Gradient and Hessian

Using the above we have for the gradient and the hessian

f = xT Ax + bT x

xf = fx

= (A + AT )x + b

2f

xxT= A + AT


2.4 Derivatives of Traces 2 DERIVATIVES

2.4 Derivatives of Traces

2.4.1 First Order

XTr(X) = I

XTr(XA) = AT (16)

XTr(AXB) = AT BT

XTr(AXT B) = BA

XTr(XT A) = A

XTr(AXT ) = A

2.4.2 Second Order

XTr(X2) = 2X

XTr(X2B) = (XB + BX)T

XTr(XT BX) = BX + BT X

XTr(XBXT ) = XBT + XB

XTr(AXBX) = AT XT BT + BT XT AT

XTr(XT X) = 2X

XTr(BXXT ) = (B + BT )X

XTr(BT XT CXB) = CT XBBT + CXBBT

XTr

[XT BXC

]= BXC + BT XCT

XTr(AXBXT C) = AT CT XBT + CAXB

XTr

[(AXb + c)(AXb + c)T

]= 2AT (AXb + c)bT

See [7].


2.5 Derivatives of Structured Matrices 2 DERIVATIVES

2.4.3 Higher Order

XTr(Xk) = k(Xk1)T

XTr(AXk) =

k1r=0

(XrAXkr1)T

XTr

[BT XT CXXT CXB

]= CXXT CXBBT

+CT XBBT XT CT X+CXBBT XT CX+CT XXT CT XBBT

2.4.4 Other

XTr(AX1B) = (X1BAX1)T = XT AT BT XT

Assume B and C to be symmetric, then

XTr

[(XT CX)1A

]= (CX(XT CX)1)(A + AT )(XT CX)1

XTr

[(XT CX)1(XT BX)

]= 2CX(XT CX)1XT BX(XT CX)1

+2BX(XT CX)1

See [7].

2.5 Derivatives of Structured Matrices

Assume that the matrix A has some structure, i.e. symmetric, toeplitz, etc.In that case the derivatives of the previous section does not apply in general.Instead, consider the following general rule for differentiating a scalar functionf(A)

df

dAij=

kl

f

Akl

AklAij

= Tr

[[f

A

]TAAij

]

The matrix differentiated with respect to itself is in this document referred toas the structure matrix of A and is defined simply by

AAij

= Sij

If A has no special structure we have simply Sij = Jij , that is, the structurematrix is simply the singleentry matrix. Many structures have a representationin singleentry matrices, see Sec. 8.2.6 for more examples of structure matrices.



2.5.1 The Chain Rule

Sometimes the objective is to find the derivative of a matrix which is a functionof another matrix. Let U = f(X), the goal is to find the derivative of thefunction g(U) with respect to X:

g(U)X

=g(f(X))

X(17)

Then the Chain Rule can then be written the following way:

g(U)X

=g(U)xij

=M

k=1

N

l=1

g(U)ukl

uklxij

(18)

Using matrix notation, this can be written as:

g(U)Xij

= Tr[(g(U)U

)TUXij

]. (19)

2.5.2 Symmetric

If A is symmetric, then Sij = Jij + Jji JijJij and therefore

df

dA=

[f

A

]+

[f

A

]T diag

[f

A

]

That is, e.g., ([5], [18]):

Tr(AX)X

= A + AT (A I), see (23) (20) det(X)

X= det(X)(2X1 (X1 I)) (21)

ln det(X)X

= 2X1 (X1 I) (22)

2.5.3 Diagonal

If X is diagonal, then ([12]):

Tr(AX)X

= A I (23)

2.5.4 Toeplitz

Like symmetric matrices and diagonal matrices also Toeplitz matrices has aspecial structure which should be taken into account when the derivative withrespect to a matrix with Toeplitz structure.



Tr(AT)T

(24)

=Tr(TA)

T

=

Tr(A) Tr([AT ]n1) Tr([[AT ]1n]n1,2) An1

Tr([AT ]1n)) Tr(A). . .

. . ....

Tr([[AT ]1n]2,n1). . .

. . .. . . Tr([[AT ]1n]n1,2)

.

.

.. . .

. . .. . . Tr([AT ]n1)

A1n Tr([[AT ]1n]2,n1) Tr([AT ]1n)) Tr(A)

(A)

As it can be seen, the derivative (A) also has a Toeplitz structure. Each valuein the diagonal is the sum of all the diagonal valued in A, the values in thediagonals next to the main diagonal equal the sum of the diagonal next to themain diagonal in AT . This result is only valid for the unconstrained Toeplitzmatrix. If the Toeplitz matrix also is symmetric, the same derivative yields

Tr(AT)T

=Tr(TA)

T= (A) + (A)T (A) I (25)


3 INVERSES

3 Inverses

3.1 Basic

3.1.1 Definition

The inverse A1 of a matrix A Cnn is defined such thatAA1 = A1A = I, (26)

where I is the nn identity matrix. If A1 exists, A is said to be nonsingular.Otherwise, A is said to be singular (see e.g. [9]).

3.1.2 Cofactors and Adjoint

The submatrix of a matrix A, denoted by [A]ij is a (n 1) (n 1) matrixobtained by deleting the ith row and the jth column of A. The (i, j) cofactorof a matrix is defined as

cof(A, i, j) = (1)i+j det([A]ij), (27)The matrix of cofactors can be created from the cofactors

cof(A) =

cof(A, 1, 1) cof(A, 1, n)

... cof(A, i, j)...

cof(A, n, 1) cof(A, n, n)

(28)

The adjoint matrix is the transpose of the cofactor matrix

adj(A) = (cof(A))T , (29)

3.1.3 Determinant

The determinant of a matrix A Cnn is defined as (see [9])

det(A) =n

j=1

(1)j+1A1j det ([A]1j)

=n

j=1

A1jcof(A, 1, j). (30)

3.1.4 Construction

The inverse matrix can be constructed, using the adjoint matrix, by

A1 =1

det(A) adj(A) (31)


3.2 Exact Relations 3 INVERSES

3.1.5 Condition number

The condition number of a matrix c(A) is the ratio between the largest and thesmallest singular value of a matrix (see Section 5.2 on singular values),

c(A) =d+d

The condition number can be used to measure how singular a matrix is. If thecondition number is large, it indicates that the matrix is nearly singular. Thecondition number can also be estimated from the matrix norms. Here

c(A) = A A1, (32)

where is a norm such as e.g the 1-norm, the 2-norm, the -norm or theFrobenius norm (see Sec 9.4 for more on matrix norms).

3.2 Exact Relations

3.2.1 The Woodbury identity

(A + CBCT )1 = A1 A1C(B1 + CT A1C)1CT A1

If P,R are positive definite, then (see [19])

(P1 + BT R1B)1BT R1 = PBT (BPBT + R)1

3.2.2 The Kailath Variant

(A + BC)1 = A1 A1B(I + CA1B)1CA1

See [4] page 153.

3.2.3 The Searle Set of Identities

The following set of identities, can be found in [15], page 151,

(I + A1)1 = A(A + I)1

(A + BBT )1B = A1B(I + BT A1B)1

(A1 + B1)1 = A(A + B)1B = B(A + B)1A

AA(A + B)1A = BB(A + B)1BA1 + B1 = A1(A + B)B1

(I + AB)1 = IA(I + BA)1B(I + AB)1A = A(I + BA)1


3.3 Implication on Inverses 3 INVERSES

3.3 Implication on Inverses

(A + B)1 = A1 + B1 AB1A = BA1BSee [15].

3.3.1 A PosDef identity

Assume P,R to be positive definite and invertible, then

(P1 + BT R1B)1BT R1 = PBT (BPBT + R)1

See [19].

3.4 Approximations

(I + A)1 = IA + A2 A3 + ...AA(I + A)1A = IA1 if A large and symmetric

If 2 is small then

(Q + 2M)1 = Q1 2Q1MQ1

3.5 Generalized Inverse

3.5.1 Definition

A generalized inverse matrix of the matrix A is any matrix A such that (see[16])

AAA = A

The matrix A is not unique.

3.6 Pseudo Inverse

3.6.1 Definition

The pseudo inverse (or Moore-Penrose inverse) of a matrix A is the matrix A+

that fulfils

I AA+A = AII A+AA+ = A+

III AA+ symmetricIV A+A symmetric

The matrix A+ is unique and does always exist.


3.6 Pseudo Inverse 3 INVERSES

3.6.2 Properties

Assume A+ to be the pseudo-inverse of A, then (See [3])

(A+)+ = A(AT )+ = (A+)T

(cA)+ = (1/c)A+

(AT A)+ = A+(AT )+

(AAT )+ = (AT )+A+

Assume A to have full rank, then

(AA+)(AA+) = AA+

(A+A)(A+A) = A+ATr(AA+) = rank(AA+) (See [16])Tr(A+A) = rank(A+A) (See [16])

3.6.3 Construction

Assume that A has full rank, then

A n n Square rank(A) = n A+ = A1A nm Broad rank(A) = n A+ = AT (AAT )1A nm Tall rank(A) = m A+ = (AT A)1AT

Assume A does not have full rank, i.e. A is nm and rank(A) = r < min(n,m).The pseudo inverse A+ can be constructed from the singular value decomposi-tion A = UDVT , by

A+ = VD+UT

A different way is this: There does always exists two matrices C n r and Dr m of rank r, such that A = CD. Using these matrices it holds that

A+ = DT (DDT )1(CT C)1CT

See [3].


4 COMPLEX MATRICES

4 Complex Matrices

4.1 Complex Derivatives

In order to differentiate an expression f(z) with respect to a complex z, theCauchy-Riemann equations have to be satisfied ([7]):

df(z)dz

=

4.1 Complex Derivatives 4 COMPLEX MATRICES

Complex Gradient Matrix: If f is a real function of a complex matrix Z,then the complex gradient matrix is given by ([2])

f(Z) = 2df(Z)dZ

(40)

=f(Z)

4.1 Complex Derivatives 4 COMPLEX MATRICES

complex number.

Tr(AXH)

5 DECOMPOSITIONS

5 Decompositions

5.1 Eigenvalues and Eigenvectors

5.1.1 Definition

The eigenvectors v and eigenvalues are the ones satisfying

Avi = ivi

AV = VD, (D)ij = ijiwhere the columns of V are the vectors vi

5.1.2 General Properties

eig(AB) = eig(BA)A is nm At most min(n,m) distinct i

rank(A) = r At most r non-zero i

5.1.3 Symmetric

Assume A is symmetric, then

VVT = I (i.e. V is orthogonal)i R (i.e. i is real)

Tr(Ap) =

ipi

eig(I + cA) = 1 + cieig(A1) = 1i

5.2 Singular Value Decomposition

Any nm matrix A can be written asA = UDVT

whereU = eigenvectors of AAT n nD =

diag(eig(AAT )) nm

V = eigenvectors of AT A mm

5.2.1 Symmetric Square decomposed into squares

Assume A to be n n and symmetric. Then[

A]

=[

V] [

D] [

VT]

where D is diagonal with the eigenvalues of A and V is orthogonal and theeigenvectors of A.


5.3 Triangular Decomposition 5 DECOMPOSITIONS

5.2.2 Square decomposed into squares

Assume A Rnn. Then[

A]

=[

V] [

D] [

UT]

where D is diagonal with the square root of the eigenvalues of AAT , V is theeigenvectors of AAT and UT is the eigenvectors of AT A.

5.2.3 Square decomposed into rectangular

Assume VDUT = 0 then we can expand the SVD of A into[

A]

=[

V V] [ D 0

0 D

] [UT

UT

]

where the SVD of A is A = VDUT .

5.2.4 Rectangular decomposition I

Assume A is nm[

A]

=[

V] [

D] [

UT]


5.2.5 Rectangular decomposition II

Assume A is nm[

A]

=[

V] D

UT

5.2.6 Rectangular decomposition III

Assume A is nm[

A]

=[

V] [

D] UT


5.3 Triangular Decomposition

5.3.1 Cholesky-decomposition

Assume A is positive definite, then

A = BT B

where B is a unique upper triangular matrix.


6 STATISTICS AND PROBABILITY

6 Statistics and Probability

6.1 Definition of Moments

Assume x Rn1 is a random variable

6.1.1 Mean

The vector of means, m, is defined by

(m)i = xi

6.1.2 Covariance

The matrix of covariance M is defined by

(M)ij = (xi xi)(xj xj)

or alternatively asM = (xm)(xm)T

6.1.3 Third moments

The matrix of third centralized moments in some contexts referred to ascoskewness is defined using the notation

m(3)ijk = (xi xi)(xj xj)(xk xk)

asM3 =

[m

(3)::1 m

(3)::2 ...m

(3)::n

]

where : denotes all elements within the given index. M3 can alternatively beexpressed as

M3 = (xm)(xm)T (xm)T

6.1.4 Fourth moments

The matrix of fourth centralized moments in some contexts referred to ascokurtosis is defined using the notation

m(4)ijkl = (xi xi)(xj xj)(xk xk)(xl xl)

asM4 =

[m

(4)::11m

(4)::21...m

(4)::n1|m(4)::12m(4)::22...m(4)::n2|...|m(4)::1nm(4)::2n...m(4)::nn

]

or alternatively as

M4 = (xm)(xm)T (xm)T (xm)T


6.2 Expectation of Linear Combinations6 STATISTICS AND PROBABILITY

6.2 Expectation of Linear Combinations

6.2.1 Linear Forms

Assume X and x to be a matrix and a vector of random variables. Then (seeSee [16])

E[AXB + C] = AE[X]B + C

Var[Ax] = AVar[x]AT

Cov[Ax,By] = ACov[x,y]BT

Assume x to be a stochastic vector with mean m, then (see [7])

E[Ax + b] = Am + bE[Ax] = Am

E[x + b] = m + b

6.2.2 Quadratic Forms

Assume A is symmetric, c = E[x] and = Var[x]. Assume also that allcoordinates xi are independent, have the same central moments 1, 2, 3, 4and denote a = diag(A). Then (See [16])

E[xT Ax] = Tr(A) + cT Ac

Var[xT Ax] = 222Tr(A2) + 42cT A2c + 43cT Aa + (4 322)aT a

Also, assume x to be a stochastic vector with mean m, and covariance M. Then(see [7])

E[(Ax + a)(Bx + b)T ] = AMBT + (Am + a)(Bm + b)T

E[xxT ] = M + mmT

E[xaT x] = (M + mmT )aE[xT axT ] = aT (M + mmT )

E[(Ax)(Ax)T ] = A(M + mmT )AT

E[(x + a)(x + a)T ] = M + (m + a)(m + a)T

E[(Ax + a)T (Bx + b)] = Tr(AMBT ) + (Am + a)T (Bm + b)E[xT x] = Tr(M) + mT m

E[xT Ax] = Tr(AM) + mT AmE[(Ax)T (Ax)] = Tr(AMAT ) + (Am)T (Am)

E[(x + a)T (x + a)] = Tr(M) + (m + a)T (m + a)

See [7].


6.3 Weighted Scalar Variable 6 STATISTICS AND PROBABILITY

6.2.3 Cubic Forms

Assume x to be a stochastic vector with independent coordinates, mean m,covariance M and central moments v3 = E[(xm)3]. Then (see [7])

E[(Ax + a)(Bx + b)T (Cx + c)] = Adiag(BT C)v3+Tr(BMCT )(Am + a)+AMCT (Bm + b)+(AMBT + (Am + a)(Bm + b)T )(Cm + c)

E[xxT x] = v3 + 2Mm + (Tr(M) + mT m)mE[(Ax + a)(Ax + a)T (Ax + a)] = Adiag(AT A)v3

+[2AMAT + (Ax + a)(Ax + a)T ](Am + a)+Tr(AMAT )(Am + a)

E[(Ax + a)bT (Cx + c)(Dx + d)T ] = (Ax + a)bT (CMDT + (Cm + c)(Dm + d)T )+(AMCT + (Am + a)(Cm + c)T )b(Dm + d)T

+bT (Cm + c)(AMDT (Am + a)(Dm + d)T )

6.3 Weighted Scalar Variable

Assume x Rn1 is a random variable, w Rn1 is a vector of constants andy is the linear combination y = wT x. Assume further that m,M2,M3,M4denotes the mean, covariance, and central third and fourth moment matrix ofthe variable x. Then it holds that

y = wT m(y y)2 = wT M2w(y y)3 = wT M3w w(y y)4 = wT M4w w w


7 GAUSSIANS

7 Gaussians

7.1 Basics

7.1.1 Density and normalization

The density of x N (m,) is

p(x) =1

det(2)exp

[1

2(xm)T 1(xm)

]

Note that if x is d-dimensional, then det(2) = (2)d det().Integration and normalization

exp

[1

2(xm)T 1(xm)

]dx =

det(2)

exp

[1

2xT Ax + bT x

]dx =

det(2A1) exp

[12bT A1b

]

exp

[1

2Tr(ST AS) + Tr(BT S)

]dS =

det(2A1) exp

[12Tr(BT A1B)

]

The derivatives of the density are

p(x)x

= p(x)1(xm)

2p

xxT= p(x)

(1(xm)(xm)T 1 1

)

7.1.2 Marginal Distribution

Assume x Nx(,) where

x =[

xaxb

] =

[ab

] =

[a cTc b

]

then

p(xa) = Nxa(a,a)p(xb) = Nxb(b,b)

7.1.3 Conditional Distribution

Assume x Nx(,) where

x =[

xaxb

] =

[ab

] =

[a cTc b

]


7.1 Basics 7 GAUSSIANS

then

p(xa|xb) = Nxa(a, a){ a = a + c1b (xb b)

a = a + c1b Tc

p(xb|xa) = Nxb(b, b){ b = b + Tc 1a (xa a)

b = b + Tc 1a c

7.1.4 Linear combination

Assume x N (mx,x) and y N (my,y) then

Ax + By + c N (Amx + Bmy + c,AxAT + ByBT )

7.1.5 Rearranging Means

NAx[m,] =

det(2(AT 1A)1)det(2)

Nx[A1m, (AT 1A)1]

7.1.6 Rearranging into squared form

If A is symmetric, then

12xT Ax + bT x = 1

2(xA1b)T A(xA1b) + 1

2bT A1b

12Tr(XT AX)+Tr(BT X) = 1

2Tr[(XA1B)T A(XA1B)]+1

2Tr(BT A1B)

7.1.7 Sum of two squared forms

In vector formulation (assuming 1,2 are symmetric)

12(xm1)T 11 (xm1)

12(xm2)T 12 (xm2)

= 12(xmc)T 1c (xmc) + C

1c = 11 +

12

mc = (11 + 12 )

1(11 m1 + 12 m2)

C =12(mT1

11 + m

T2

12 )(

11 +

12 )

1(11 m1 + 12 m2)

12

(mT1

11 m1 + m

T2

12 m2

)


7.2 Moments 7 GAUSSIANS

In a trace formulation (assuming 1,2 are symmetric)

12Tr((XM1)T 11 (XM1))

12Tr((XM2)T 12 (XM2))

= 12Tr[(XMc)T 1c (XMc)] + C

1c = 11 +

12

Mc = (11 + 12 )

1(11 M1 + 12 M2)

C =12Tr

[(11 M1 +

12 M2)

T (11 + 12 )

1(11 M1 + 12 M2)

]

12Tr(MT1

11 M1 + M

T2

12 M2)

7.1.8 Product of gaussian densities

Let Nx(m,) denote a density of x, thenNx(m1,1) Nx(m2,2) = ccNx(mc,c)

cc = Nm1(m2, (1 + 2))=

1det(2(1 + 2))

exp[1

2(m1 m2)T (1 + 2)1(m1 m2)

]

mc = (11 + 12 )

1(11 m1 + 12 m2)

c = (11 + 12 )

1

but note that the product is not normalized as a density of x.

7.2 Moments

7.2.1 Mean and covariance of linear forms

First and second moments. Assume x N (m,)E(x) = m

Cov(x,x) = Var(x) = = E(xxT ) E(x)E(xT ) = E(xxT )mmT

As for any other distribution is holds for gaussians that

E[Ax] = AE[x]

Var[Ax] = AVar[x]AT

Cov[Ax,By] = ACov[x,y]BT


7.2 Moments 7 GAUSSIANS

7.2.2 Mean and variance of square forms

Mean and variance of square forms: Assume x N (m,)

E(xxT ) = + mmT

E[xT Ax] = Tr(A) + mT AmVar(xT Ax) = 24Tr(A2) + 42mT A2m

E[(xm)T A(xm)] = (mm)T A(mm) + Tr(A)

Assume x N (0, 2I) and A and B to be symmetric, then

Cov(xT Ax,xT Bx) = 24Tr(AB)

7.2.3 Cubic forms

E[xbT xxT ] = mbT (M + mmT ) + (M + mmT )bmT

+bT m(MmmT )

7.2.4 Mean of Quartic Forms

E[xxT xxT ] = 2( + mmT )2 + mT m(mmT )+Tr()( + mmT )

E[xxT AxxT ] = ( + mmT )(A + AT )( + mmT )+mT Am(mmT ) + Tr[A( + mmT )]

E[xT xxT x] = 2Tr(2) + 4mT m + (Tr() + mT m)2

E[xT AxxT Bx] = Tr[A(B + BT )] + mT (A + AT )(B + BT )m+(Tr(A) + mT Am)(Tr(B) + mT Bm)

E[aT xbT xcT xdT x]= (aT ( + mmT )b)(cT ( + mmT )d)

+(aT ( + mmT )c)(bT ( + mmT )d)+(aT ( + mmT )d)(bT ( + mmT )c) 2aT mbT mcT mdT m

E[(Ax + a)(Bx + b)T (Cx + c)(Dx + d)T ]= [ABT + (Am + a)(Bm + b)T ][CDT + (Cm + c)(Dm + d)T ]

+[ACT + (Am + a)(Cm + c)T ][BDT + (Bm + b)(Dm + d)T ]+(Bm + b)T (Cm + c)[ADT (Am + a)(Dm + d)T ]+Tr(BCT )[ADT + (Am + a)(Dm + d)T ]


7.3 Miscellaneous 7 GAUSSIANS

E[(Ax + a)T (Bx + b)(Cx + c)T (Dx + d)]= Tr[A(CT D + DT C)BT ]

+[(Am + a)T B + (Bm + b)T A][CT (Dm + d) + DT (Cm + c)]+[Tr(ABT ) + (Am + a)T (Bm + b)][Tr(CDT ) + (Cm + c)T (Dm + d)]

See [7].

7.2.5 Moments

E[x] =

k

kmk

Cov(x) =

k

kkk(k + mkmTk mkmTk)

7.3 Miscellaneous

7.3.1 Whitening

Assume x N (m,) then

z = 1/2(xm) N (0, I)

Conversely having z N (0, I) one can generate data x N (m,) by setting

x = 1/2z + m N (m,)

Note that 1/2 means the matrix which fulfils 1/21/2 = , and that it existsand is unique since is positive definite.

7.3.2 The Chi-Square connection

Assume x N (m,) and x to be n dimensional, then

z = (xm)T 1(xm) 2n

7.3.3 Entropy

Entropy of a D-dimensional gaussian

H(x) =N (m,) lnN (m,)dx = ln

det(2) D

2


7.4 Mixture of Gaussians 7 GAUSSIANS

7.4 Mixture of Gaussians

7.4.1 Density

The variable x is distributed as a mixture of gaussians if it has the density

p(x) =K

k=1

k1

det(2k)exp

[1

2(xmk)T 1k (xmk)

]

where k sum to 1 and the k all are positive definite.

7.4.2 Derivatives

Defining p(s) =

k kNs(k,k) one get

ln p(s)j

=jNs(j ,j)k kNs(k,k)

jln[jNs(j ,j)]


1j

ln p(s)j


jln[jNs(j ,j)]


[1k (s k)]

ln p(s)j


jln[jNs(j ,j)]


12

[Tj +

Tj (s j)(s j)T Tj

]

But k and k needs to be constrained.


8 SPECIAL MATRICES

8 Special Matrices

8.1 Units, Permutation and Shift

8.1.1 Unit vector

Let ei Rn1 be the ith unit vector, i.e. the vector which is zero in all entriesexcept the ith at which it is 1.

8.1.2 Rows and Columns

i.th row of A = eTi Aj.th column of A = Aej

8.1.3 Permutations

Let P be some permutation matrix, e.g.

P =

0 1 01 0 00 0 1

= [ e2 e1 e3

]=

eT2eT1eT3

then

AP =[

Ae2 Ae1 Ae2]

PA =

eT2 AeT1 AeT3 A

That is, the first is a matrix which has columns of A but in permuted sequenceand the second is a matrix which has the rows of A but in the permuted se-quence.

8.1.4 Translation, Shift or Lag Operators

Let L denote the lag (or translation or shift) operator defined on a 4 4example by

L =

0 0 0 01 0 0 00 1 0 00 0 1 0

i.e. a matrix of zeros with one on the sub-diagonal, (L)ij = i,j+1. With somesignal xt for t = 1, ..., N , the n.th power of the lag operator shifts the indices,i.e.

(Lnx)t ={ 0 for t = 1, .., n

xtn for t = n + 1, ..., N


8.2 The Singleentry Matrix 8 SPECIAL MATRICES

A related but slightly different matrix is the recurrent shifted operator definedon a 4x4 example by

L =

0 0 0 11 0 0 00 1 0 00 0 1 0

i.e. a matrix defined by (L)ij = i,j+1 + i,1j,dim(L). On a signal x it has theeffect

(Lnx)t = xt , t = [(t n) mod N ] + 1That is, L is like the shift operator L except that it wraps the signal as if itwas periodic and shifted (substituting the zeros with the rear end of the signal).Note that L is invertible and orthogonal, i.e.

L1 = LT

8.2 The Singleentry Matrix

8.2.1 Definition

The single-entry matrix Jij Rnn is defined as the matrix which is zeroeverywhere except in the entry (i, j) in which it is 1. In a 4 4 example onemight have

J23 =

0 0 0 00 0 1 00 0 0 00 0 0 0

The single-entry matrix is very useful when working with derivatives of expres-sions involving matrices.

8.2.2 Swap and Zeros

Assume A to be nm and Jij to be m p

AJij =[

0 0 . . . Ai . . . 0]

i.e. an n p matrix of zeros with the i.th column of A in place of the j.thcolumn. Assume A to be nm and Jij to be p n

JijA =

0...0Aj0...0


8.2 The Singleentry Matrix 8 SPECIAL MATRICES

i.e. an p m matrix of zeros with the j.th row of A in the placed of the i.throw.

8.2.3 Rewriting product of elements

AkiBjl = (AeieTj B)kl = (AJijB)kl

AikBlj = (AT eieTj BT )kl = (AT JijBT )kl

AikBjl = (AT eieTj B)kl = (AT JijB)kl

AkiBlj = (AeieTj BT )kl = (AJijBT )kl

8.2.4 Properties of the Singleentry Matrix

If i = jJijJij = Jij (Jij)T (Jij)T = Jij

Jij(Jij)T = Jij (Jij)T Jij = Jij

If i 6= jJijJij = 0 (Jij)T (Jij)T = 0

Jij(Jij)T = Jii (Jij)T Jij = Jjj

8.2.5 The Singleentry Matrix in Scalar Expressions

Assume A is nm and J is m n, then

Tr(AJij) = Tr(JijA) = (AT )ij

Assume A is n n, J is nm and B is m n, then

Tr(AJijB) = (AT BT )ij

Tr(AJjiB) = (BA)ij

Tr(AJijJijB) = diag(AT BT )ij

Assume A is n n, Jij is nm B is m n, then

xT AJijBx = (AT xxT BT )ij

xT AJijJijBx = diag(AT xxT BT )ij


8.3 Symmetric and Antisymmetric 8 SPECIAL MATRICES

8.2.6 Structure Matrices

The structure matrix is defined by

AAij

= Sij

If A has no special structure then

Sij = Jij

If A is symmetric thenSij = Jij + Jji JijJij

8.3 Symmetric and Antisymmetric

8.3.1 Symmetric

The matrix A is said to be symmetric if

A = AT

Symmetric matrices have many important properties, e.g. that their eigenvaluesare real and eigenvalues orthogonal.

8.3.2 Antisymmetric

The antisymmetric matrix is also known as the skew symmetric matrix. It hasthe following property from which it is defined

A = AT

Hereby, it can be seen that the antisymmetric matrices always have a zerodiagonal. The n n antisymmetric matrices also have the following properties.

det(AT ) = det(A) = (1)n det(A) det(A) = det(A) = 0, if n is odd

8.4 Toeplitz Matrices

A Toeplitz matrix T is a matrix where the elements of each diagonal is thesame. In the n n square case, it has the following structure:

T =

t11 t12 t1nt21

. . . . . ....

.... . . . . . t12

tn1 t21 t11

=

t0 t1 tn1t1

. . . . . ....

.... . . . . . t1

t(n1) t1 t0

(58)


8.4 Toeplitz Matrices 8 SPECIAL MATRICES

A Toeplitz matrix is persymmetric. If a matrix is persymmetric (or orthosym-metric), it means that the matrix is symmetric about its northeast-southwestdiagonal (anti-diagonal) [9]. Persymmetric matrices is a larger class of matrices,since a persymmetric matrix not necessarily has a Toeplitz structure. There aresome special cases of Toeplitz matrices. The symmetric Toeplitz matrix is givenby:

T =

t0 t1 tn1t1

. . . . . ....

.... . . . . . t1

t(n1) t1 t0

(59)

The circular Toeplitz matrix:

TC =

t0 t1 tn1tn

. . . . . ....

.... . . . . . t1

t1 tn1 t0

(60)

The upper triangular Toeplitz matrix:

TU =

t0 t1 tn10

. . . . . ....

.... . . . . . t1

0 0 t0

, (61)

and the lower triangular Toeplitz matrix:

TL =

t0 0 0t1

. . . . . ....

.... . . . . . 0

t(n1) t1 t0

(62)

8.4.1 Properties of Toeplitz Matrices

The Toeplitz matrix has some computational advantages. The addition of twoToeplitz matrices can be done with O(n) flops, multiplication of two Toeplitzmatrices can be done in O(n ln n) flops. Toeplitz equation systems can besolved in O(n2) flops. The inverse of a positive definite Toeplitz matrix canbe found in O(n2) flops too. The inverse of a Toeplitz matrix is persymmetric.The product of two lower triangular Toeplitz matrices is a Toeplitz matrix.The following important relation between the circulant matrix and the discreteFourier transform (DFT) exists

TC = F1n (I (Fnt))Fn, (63)


8.5 Positive Definite and Semi-definite Matrices 8 SPECIAL MATRICES

where t = [t0, t1, , tn1]T is the first row of TC . Fn is the n n discreteFourier transform (DFT) matrix defined as

Fn = DFT(I). (64)

Likewise, F1n is the inverse DFT matrix defined as

F1n = IDFT(I) = (DFT(I))1. (65)

More information on Toeplitz matrices and circulant matrices can be found in[10, 7].

8.5 Positive Definite and Semi-definite Matrices

8.5.1 Definitions

A matrix A is positive definite if and only if

xT Ax > 0, x

A matrix A is positive semi-definite if and only if

xT Ax 0, x

Note that if A is positive definite, then A is also positive semi-definite.

8.5.2 Eigenvalues

The following holds with respect to the eigenvalues:

A pos. def. eig(A) > 0A pos. semi-def. eig(A) 0

8.5.3 Trace

The following holds with respect to the trace:

A pos. def. Tr(A) > 0A pos. semi-def. Tr(A) 0

8.5.4 Inverse

If A is positive definite, then A is invertible and A1 is also positive definite.

8.5.5 Diagonal

If A is positive definite, then Aii > 0,i


8.6 Block matrices 8 SPECIAL MATRICES

8.5.6 Decomposition I

The matrix A is positive semi-definite of rank r there exists a matrix B ofrank r such that A = BBT

The matrix A is positive definite there exists an invertible matrix B suchthat A = BBT

8.5.7 Decomposition II

Assume A is an n n positive semi-definite, then there exists an n r matrixB of rank r such that BT AB = I.

8.5.8 Equation with zeros

Assume A is positive semi-definite, then XT AX = 0 AX = 0

8.5.9 Rank of product

Assume A is positive definite, then rank(BABT ) = rank(B)

8.5.10 Positive definite property

If A is n n positive definite and B is r n of rank r, then BABT is positivedefinite.

8.5.11 Outer Product

If X is n r of rank r, then XXT is positive definite.

8.5.12 Small pertubations

If A is positive definite and B is symmetric, then A tB is positive definite forsufficiently small t.

8.6 Block matrices

Let Aij denote the ijth block of A.

8.6.1 Multiplication

Assuming the dimensions of the blocks matches we have[

A11 A12A21 A22

] [B11 B12B21 B22

]=

[A11B11 + A12B21 A11B12 + A12B22A21B11 + A22B21 A21B12 + A22B22

]


8.6 Block matrices 8 SPECIAL MATRICES

8.6.2 The Determinant

The determinant can be expressed as by the use of

C1 = A11 A12A122 A21C2 = A22 A21A111 A12

as

det([

A11 A12A21 A22

])= det(A22) det(C1) = det(A11) det(C2)

8.6.3 The Inverse

The inverse can be expressed as by the use of

C1 = A11 A12A122 A21C2 = A22 A21A111 A12

as [A11 A12A21 A22

]1=

[C11 A111 A12C12

C12 A21A111 C12

]

=[

A111 + A111 A12C

12 A21A

111 C11 A12A122

A122 A21C11 A122 + A122 A21C11 A12A122

]

8.6.4 Block diagonal

For block diagonal matrices we have

[A11 00 A22

]1=

[(A11)1 0

0 (A22)1

]

det([

A11 00 A22

])= det(A11) det(A22)


9 FUNCTIONS AND OPERATORS

9 Functions and Operators

9.1 Functions and Series

9.1.1 Finite Series

(Xn I)(X I)1 = I + X + X2 + ... + Xn1

9.1.2 Taylor Expansion of Scalar Function

Consider some scalar function f(x) which takes the vector x as an argument.This we can Taylor expand around x0

f(x) = f(x0) + g(x0)T (x x0) + 12(x x0)T H(x0)(x x0)

where

g(x0) =f(x)

x

x0

H(x0) =2f(x)xxT

x0

9.1.3 Matrix Functions by Infinite Series

As for analytical functions in one dimension, one can define a matrix functionfor square matrices X by an infinite series

f(X) =

n=0

cnXn

assuming the limit exists and is finite. If the coefficients cn fulfils

n cnxn < ,

then one can prove that the above series exists and is finite, see [1]. Thus forany analytical function f(x) there exists a corresponding matrix function f(x)constructed by the Taylor expansion. Using this one can prove the followingresults:1) A matrix A is a zero of its own characteristic polynomium [1]:

p() = det(IA) =

n

cnn p(A) = 0

2) If A is square it holds that [1]

A = UBU1 f(A) = Uf(B)U1

3) A useful fact when using power series is that

An 0 for n if |A| < 1


9.2 Kronecker and Vec Operator 9 FUNCTIONS AND OPERATORS

9.1.4 Exponential Matrix Function

In analogy to the ordinary scalar exponential function, one can define exponen-tial and logarithmic matrix functions:

eA

n=0

1n!

An = I + A +12A2 + ...

eA

n=0

1n!

(1)nAn = IA + 12A2 ...

etA

n=0

1n!

(tA)n = I + tA +12t2A2 + ...

ln(I + A)

n=1

(1)n1n

An = A 12A2 +

13A3 ...

Some of the properties of the exponential function are [1]

eAeB = eA+B if AB = BA

(eA)1 = eA

d

dtetA = AetA, t R

9.1.5 Trigonometric Functions

sin(A)

n=0

(1)nA2n+1(2n + 1)!

= A 13!

A3 +15!

A5 ...

cos(A)

n=0

(1)nA2n(2n)!

= I 12!

A2 +14!

A4 ...

9.2 Kronecker and Vec Operator

9.2.1 The Kronecker Product

The Kronecker product of an m n matrix A and an r q matrix B, is anmr nq matrix, AB defined as

AB =

A11B A12B ... A1nBA21B A22B ... A2nB

......

Am1B Am2B ... AmnB


9.3 Solutions to Systems of Equations 9 FUNCTIONS AND OPERATORS

The Kronecker product has the following properties (see [12])

A (B + C) = AB + ACAB 6= BA

A (BC) = (AB)C(AA BB) = AB(AB)

(AB)T = AT BT(AB)(CD) = ABCD

(AB)1 = A1 B1rank(AB) = rank(A)rank(B)

Tr(AB) = Tr(A)Tr(B)det(AB) = det(A)rank(B) det(B)rank(A)

9.2.2 The Vec Operator

The vec-operator applied on a matrix A stacks the columns into a vector, i.e.for a 2 2 matrix

A =[

A11 A12A21 A22

]vec(A) =

A11A21A12A22

Properties of the vec-operator include (see [12])

vec(AXB) = (BT A)vec(X)Tr(AT B) = vec(A)T vec(B)

vec(A + B) = vec(A) + vec(B)vec(A) = vec(A)

9.3 Solutions to Systems of Equations

9.3.1 Existence in Linear Systems

Assume A is nm and consider the linear system

Ax = b

Construct the augmented matrix B = [A b] then

Condition Solutionrank(A) = rank(B) = m Unique solution xrank(A) = rank(B) < m Many solutions xrank(A) < rank(B) No solutions x


9.3 Solutions to Systems of Equations 9 FUNCTIONS AND OPERATORS

9.3.2 Standard Square

Assume A is square and invertible, then

Ax = b x = A1b

9.3.3 Degenerated Square

9.3.4 Over-determined Rectangular

Assume A to be nm, n > m (tall) and rank(A) = m, then

Ax = b x = (AT A)1AT b = A+b

that is if there exists a solution x at all! If there is no solution the followingcan be useful:

Ax = b xmin = A+bNow xmin is the vector x which minimizes ||Ax b||2, i.e. the vector which isleast wrong. The matrix A+ is the pseudo-inverse of A. See [3].

9.3.5 Under-determined Rectangular

Assume A is nm and n < m (broad).

Ax = b xmin = AT (AAT )1b

The equation have many solutions x. But xmin is the solution which minimizes||Axb||2 and also the solution with the smallest norm ||x||2. The same holdsfor a matrix version: Assume A is nm, X is m n and B is n n, then

AX = B Xmin = A+B

The equation have many solutions X. But Xmin is the solution which minimizes||AXB||2 and also the solution with the smallest norm ||X||2. See [3].

Similar but different: Assume A is square n n and the matrices B0,B1are nN , where N > n, then if B0 has maximal rank

AB0 = B1 Amin = B1BT0 (B0BT0 )1

where Amin denotes the matrix which is optimal in a least square sense. Aninterpretation is that A is the linear approximation which maps the columnsvectors of B0 into the columns vectors of B1.

9.3.6 Linear form and zeros

Ax = 0, x A = 0


9.4 Matrix Norms 9 FUNCTIONS AND OPERATORS

9.3.7 Square form and zeros

If A is symmetric, then

xT Ax = 0, x A = 0

9.3.8 The Lyapunov Equation

AX + XB = C

vec(X) = (IA + BT I)1vec(C)See Sec 9.2.1 and 9.2.2 for details on the Kronecker product and the vec operator.

9.3.9 Encapsulating Sum

nAnXBn = C

vec(X) =(

nBTn An

)1vec(C)

See Sec 9.2.1 and 9.2.2 for details on the Kronecker product and the vec operator.

9.4 Matrix Norms

9.4.1 Definitions

A matrix norm is a mapping which fulfils

||A|| 0 ||A|| = 0 A = 0||cA|| = |c|||A||, c R

||A + B|| ||A||+ ||B||

9.4.2 Examples

||A||1 = maxj

i |Aij |||A||2 =

max eig(AT A)

||A||p = (max||x||p=1 ||Ax||p)1/p||A|| = maxi

j |Aij |

||A||F =

ij |Aij |2 =

Tr(AAH) (Frobenius)

||A||max = maxij |Aij |||A||KF = ||sing(A)||1 (Ky Fan)

where sing(A) is the vector of singular values of the matrix A.


9.5 Rank 9 FUNCTIONS AND OPERATORS

9.4.3 Inequalities

E. H. Rasmussen has in yet unpublished material derived and collected thefollowing inequalities. They are collected in a table as below, assuming A is anm n, and d = min{m, n}

||A||max ||A||1 ||A|| ||A||2 ||A||F ||A||KF||A||max 1 1 1 1 1||A||1 m m

m

m

m

||A|| n n

n

n

n||A||2

mn

n

m 1 1

||A||F

mn

n

m

d 1||A||KF

mnd

nd

md d

d

which are to be read as, e.g.

||A||2

m ||A||

9.5 Rank

9.5.1 Sylvesters Inequality

If A is m n and B is n r, thenrank(A) + rank(B) n rank(AB) min{rank(A), rank(B)}

9.6 Integral Involving Dirac Delta Functions

Assuming A to be square, then

p(s)(xAs)ds = 1det(A)

p(A1x)

Assuming A to be underdetermined, i.e. tall, then

p(s)(xAs)ds ={

1det(AT A)

p(A+x) if x = AA+x

0 elsewhere

}

See [8].

9.7 Miscellaneous

For any A it holds that

rank(A) = rank(AT ) = rank(AAT ) = rank(AT A)

It holds that

A is positive definite B invertible, such that A = BBT

9.7.1 Orthogonal matrix

If A is orthogonal, then det(A) = 1.


A ONE-DIMENSIONAL RESULTS

A One-dimensional Results

A.1 Gaussian

A.1.1 Density

p(x) =1

22exp

( (x )

2

22

)

A.1.2 Normalization

e

(s)222 ds =

22

e(ax

2+bx+c)dx =

aexp

[b2 4ac

4a

]

ec2x

2+c1x+c0dx =

c2 exp[c21 4c2c04c2

]

A.1.3 Derivatives

p(x)

= p(x)(x )

2

ln p(x)

=(x )

2

p(x)

= p(x)1

[(x )2

2 1

]

ln p(x)

=1

[(x )2

2 1

]

A.1.4 Completing the Squares

c2x2 + c1x + c0 = a(x b)2 + w

a = c2 b = 12c1c2

w =14

c21c2

+ c0

orc2x

2 + c1x + c0 = 122 (x )2 + d

=c12c2

2 =12c2

d = c0 c21

4c2


A.2 One Dimensional Mixture of GaussiansA ONE-DIMENSIONAL RESULTS

A.1.5 Moments

If the density is expressed by

p(x) =1

22exp

[ (s )

2

22

]or p(x) = C exp(c2x2 + c1x)

then the first few basic moments are

x = = c12c2x2 = 2 + 2 = 12c2 +

(c12c2

)2

x3 = 32 + 3 = c1(2c2)2[3 c212c2

]

x4 = 4 + 622 + 34 =(

c12c2

)4+ 6

(c12c2

)2 (12c2

)+ 3

(1

2c2

)2

and the central moments are

(x ) = 0 = 0(x )2 = 2 =

[12c2

]

(x )3 = 0 = 0(x )4 = 34 = 3

[1

2c2

]2

A kind of pseudo-moments (un-normalized integrals) can easily be derived as

exp(c2x2 + c1x)xndx = Zxn =

c2 exp[

c214c2

]xn

From the un-centralized moments one can derive other entities like

x2 x2 = 2 = 12c2x3 x2x = 22 = 2c1(2c2)2x4 x22 = 24 + 422 = 2(2c2)2

[1 4 c212c2

]

A.2 One Dimensional Mixture of Gaussians

A.2.1 Density and Normalization

p(s) =K

k

k22k

exp[1

2(s k)2

2k

]

A.2.2 Moments

An useful fact of MoG, is that

xn =

k

kxnk


A.2 One Dimensional Mixture of GaussiansA ONE-DIMENSIONAL RESULTS

where k denotes average with respect to the k.th component. We can calculatethe first four moments from the densities

p(x) =

k

k1

22kexp

[1

2(x k)2

2k

]

p(x) =

k

kCk exp[ck2x

2 + ck1x]

as

x = k kk =

k k

[ck12ck2

]

x2 = k k(2k + 2k) =

k k

[12ck2

+(ck12ck2

)2]

x3 = k k(32kk + 3k) =

k k

[ck1

(2ck2)2

[3 c2k12ck2

]]

x4 = k k(4k + 62k2k + 34k) =

k k

[(1

2ck2

)2 [(ck12ck2

)2 6 c2k12ck2 + 3

]]

If all the gaussians are centered, i.e. k = 0 for all k, then

x = 0 = 0x2 = k k2k =

k k

[12ck2

]

x3 = 0 = 0x4 = k k34k =

k k3

[12ck2

]2

From the un-centralized moments one can derive other entities like

x2 x2 = k,k kk[2k +

2k kk

]x3 x2x = k,k kk

[32kk +

3k (2k + 2k)k

]x4 x22 = k,k kk

[4k + 6

2k

2k + 3

4k (2k + 2k)(2k + 2k)

]

A.2.3 Derivatives

Defining p(s) =

k kNs(k, 2k) we get for a parameter j of the j.th compo-nent

ln p(s)j

=jNs(j , 2j )k kNs(k, 2k)

ln(jNs(j , 2j ))j

that is,

ln p(s)j


1j

ln p(s)j


(s j)2j

ln p(s)j


1j

[(s j)2

2j 1

]


B PROOFS AND DETAILS

Note that k must be constrained to be proper ratios. Defining the ratios byj = erj /

k e

rk , we obtain

ln p(s)rj

=

l

ln p(s)l

lrj

wherelrj

= l(lj j)

B Proofs and Details

B.1 Misc Proofs

B.1.1 Proof of Equation 14

Essentially we need to calculate

(Xn)klXij

=

Xij

u1,...,un1

Xk,u1Xu1,u2 ...Xun1,l

= k,iu1,jXu1,u2 ...Xun1,l+Xk,u1u1,iu2,j ...Xun1,l

...+Xk,u1Xu1,u2 ...un1,il,j

=n1r=0

(Xr)ki(Xn1r)jl

=n1r=0

(XrJijXn1r)kl

Using the properties of the single entry matrix found in Sec. 8.2.4, the resultfollows easily.

B.1.2 Details on Eq. 67

det(XHAX) = det(XHAX)Tr[(XHAX)1(XHAX)]= det(XHAX)Tr[(XHAX)1((XH)AX + XH(AX))]= det(XHAX)

(Tr[(XHAX)1(XH)AX]

+Tr[(XHAX)1XH(AX)])

= det(XHAX)(Tr[AX(XHAX)1(XH)]

+Tr[(XHAX)1XHA(X)])


B.1 Misc Proofs B PROOFS AND DETAILS

First, the derivative is found with respect to the real part of X

det(XHAX)

REFERENCES REFERENCES

References

[1] Karl Gustav Andersson and Lars-Christer Boiers. Ordinaera differentialek-vationer. Studenterlitteratur, 1992.

[2] Jorn Anemuller, Terrence J. Sejnowski, and Scott Makeig. Complex inde-pendent component analysis of frequency-domain electroencephalographicdata. Neural Networks, 16(9):13111323, November 2003.

[3] S. Barnet. Matrices. Methods and Applications. Oxford Applied Mathe-matics and Computin Science Series. Clarendon Press, 1990.

[4] Christoffer Bishop. Neural Networks for Pattern Recognition. Oxford Uni-versity Press, 1995.

[5] Robert J. Boik. Lecture notes: Statistics 550. Online, April 22 2002. Notes.

[6] D. H. Brandwood. A complex gradient operator and its application inadaptive array theory. IEE Proceedings, 130(1):1116, February 1983. PTS.F and H.

[7] M. Brookes. Matrix Reference Manual, 2004. Website May 20, 2004.

[8] Mads Dyrholm. Some matrix results, 2004. Website August 23, 2004.

[9] Gene H. Golub and Charles F. van Loan. Matrix Computations. The JohnsHopkins University Press, Baltimore, 3rd edition, 1996.

[10] Robert M. Gray. Toeplitz and circulant matrices: A review. Technicalreport, Information Systems Laboratory, Department of Electrical Engi-neering,Stanford University, Stanford, California 94305, August 2002.

[11] Simon Haykin. Adaptive Filter Theory. Prentice Hall, Upper Saddle River,NJ, 4th edition, 2002.

[12] Thomas P. Minka. Old and new matrix algebra useful for statistics, De-cember 2000. Notes.

[13] L. Parra and C. Spence. Convolutive blind separation of non-stationarysources. In IEEE Transactions Speech and Audio Processing, pages 320327, May 2000.

[14] Laurent Schwartz. Cours dAnalyse, volume II. Hermann, Paris, 1967. Asreferenced in [11].

[15] Shayle R. Searle. Matrix Algebra Useful for Statistics. John Wiley andSons, 1982.

[16] G. Seber and A. Lee. Linear Regression Analysis. John Wiley and Sons,2002.

[17] S. M. Selby. Standard Mathematical Tables. CRC Press, 1974.


REFERENCES REFERENCES

[18] Inna Stainvas. Matrix algebra in differential calculus. Neural ComputingResearch Group, Information Engeneering, Aston University, UK, August2002. Notes.

[19] Max Welling. The kalman filter. Lecture Note.


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