Cookbook Matrix

The Matrix Cookbook

Kaare Brandt PetersenMichael Syskind Pedersen

Version: January 5, 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 . . . . . . . . . . . . . . . . . . . . . . . . . 6

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 143.1 Exact Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Implication on Inverses . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Pseudo Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Complex Matrices 174.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Decompositions 205.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 205.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . 205.3 Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . . 21

6 General Statistics and Probability 226.1 Moments of any distribution . . . . . . . . . . . . . . . . . . . . . 226.2 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Gaussians 247.1 One Dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.5 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 297.6 Mixture of Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Miscellaneous 318.1 Functions and Series . . . . . . . . . . . . . . . . . . . . . . . . . 318.2 Indices, Entries and Vectors . . . . . . . . . . . . . . . . . . . . . 328.3 Solutions to Systems of Equations . . . . . . . . . . . . . . . . . 358.4 Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.5 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.6 Positive Definite and Semi-definite Matrices . . . . . . . . . . . . 38

Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 2

CONTENTS CONTENTS

8.7 Integral Involving Dirac Delta Functions . . . . . . . . . . . . . . 398.8 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A Proofs and Details 41


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 AA1/2 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 Aa 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 = BTAT

(ABC...)T = ...CTBTAT

(AH)1 = (A1)H

(A+B)H = AH +BH

(AB)H = BHAH

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

1.1 Trace and Determinants

Tr(A) =i

Aii =i

i, 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) =i

i i = eig(A)

det(AB) = det(A) det(B), if A and B are invertible

det(A1) =1

det(A)


1.2 The Special Case 2x2 1 BASICS

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 =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 ([10]):

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


2.2 Derivatives of an Inverse 2 DERIVATIVES

2.1.3 Square forms

If X is square and invertible, then

det(XTAX)X

= 2det(XTAX)XT

If X is not square but A is symmetric, then

det(XTAX)X

= 2det(XTAX)AX(XTAX)1

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

det(XTAX)X

= det(XTAX)(AX(XTAX)1 +ATX(XTATX)1) (13)

2.1.4 Other nonlinear forms

Some special cases are (See [8])

ln det(XTX)|X

= 2(X+)T

ln det(XTX)X+

= 2XT

ln | det(X)|X

= (X1)T = (XT )1

det(Xk)X

= k det(Xk)XT

See [7].

2.2 Derivatives of an Inverse

From [15] we have the basic identity

Y1

x= Y1 Y

xY1

from which it follows

(X1)klXij

= (X1)ki(X1)jl

aTX1bX

= XTabTXT

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

xTbx

=bTxx

= b

aTXbX

= abT

aTXTbX

= baT

aTXaX

=aTXTa

X= aaT

XXij

= Jij

(XA)ijXmn

= im(A)nj = (JmnA)ij

(XTA)ijXmn

= in(A)mj = (JnmA)ij

2.3.2 Second Order

Xij

klmn

XklXmn = 2kl

Xkl

bTXTXcX

= X(bcT + cbT )

(Bx+ b)TC(Dx+ d)x

= BTC(Dx+ d) +DTCT (Bx+ b)

(XTBX)klXij

= lj(XTB)ki + kj(BX)il

(XTBX)Xij

= XTBJij + JjiBX (Jij)kl = ikjl

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

xTBxx

= (B+BT )x

bTXTDXcX

= DTXbcT +DXcbT

X(Xb+ c)TD(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)TW(xAs) = 2ATW(xAs)

x(xAs)TW(xAs) = 2W(xAs)

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

2.3.3 Higher order and non-linear

XaTXnb =

n1r=0

(Xr)TabT (Xn1r)T (14)

XaT (Xn)TXnb =

n1r=0

[Xn1rabT (Xn)TXr

+(Xr)TXnabT (Xn1r)T]

(15)

See A.0.1 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

xsTAs =

[sx

]T(A+AT )s

xsTAr =

[sx

]TAs+

[rx

]TAT r

2.3.4 Gradient and Hessian

Using the above we have for the gradient and the hessian

f = xTAx+ bTx

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(XB) = BT (16)

XTr(BXC) = BTCT

XTr(BXTC) = CB

XTr(XTC) = C

XTr(BXT ) = B

2.4.2 Second Order

XTr(X2) = 2X

XTr(X2B) = (XB+BX)T

XTr(XTBX) = BX+BTX

XTr(XBXT ) = XBT +XB

XTr(XTX) = 2X

XTr(BXXT ) = (B+BT )X

XTr(BTXTCXB) = CTXBBT +CXBBT

XTr[XTBXC

]= BXC+BTXCT

XTr(AXBXTC) = ATCTXBT +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[BTXTCXXTCXB

]= CXXTCXBBT +CTXBBTXTCTX

+ CXBBTXTCX+CTXXTCTXBBT

2.4.4 Other

XTr(AX1B) = (X1BAX1)T = XTATBTXT

Assume B and C to be symmetric, then

XTr[(XTCX)1A

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

XTr[(XTCX)1(XTBX)

]= 2CX(XTCX)1XTBX(XTCX)1

+2BX(XTCX)1

See [7].

2.5 Derivatives of Structured Matrices

Assume that the matrix A has some structure, i.e. is symmetric, toeplitz, etc.In that case the derivatives of the previous section does not apply in general.In stead, 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. 8.2.7 for more examples of structure matrices.


2.5 Derivatives of Structured Matrices 2 DERIVATIVES

2.5.1 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], [16]):

Tr(AX)X

= A+AT (A I), see (20) (17) det(X)X

= 2X (X I) (18) ln det(X)

X= 2X1 (X1 I) (19)

2.5.2 Diagonal

If X is diagonal, then ([10]):

Tr(AX)X

= A I (20)


3 INVERSES

3 Inverses

3.1 Exact Relations

3.1.1 The Woodbury identity

(A+CBCT )1 = A1 A1C(B1 +CTA1C)1CTA1

If P,R are positive definite, then (see [17])

(P1 +BTR1B)1BTR1 = PBT (BPBT +R)1

3.1.2 The Kailath Variant

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

See [4] page 153.

3.1.3 The Searle Set of Identities

The following set of identities, can be found in [13], page 151,

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

(A+BBT )1B = A1B(I+BTA1B)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+AB)1

3.2 Implication on Inverses

(A+B)1 = A1 +B1 AB1A = BA1BSee [13].

3.2.1 A PosDef identity

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

(P1 +BTR1B)1BTR1 = PBT (BPBT +R)1

See [?].


3.3 Approximations 3 INVERSES

3.3 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

3.4 Generalized Inverse

3.4.1 Definition

A generalized inverse matrix of the matrix A is any matrix A such that

AAA = A

The matrix A is not unique.

3.5 Pseudo Inverse

3.5.1 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.

3.5.2 Properties

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

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

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

(ATA)+ = 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 [14])Tr(A+A) = rank(A+A) (See [14])


3.5 Pseudo Inverse 3 INVERSES

3.5.3 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+ = (ATA)1AT

AssumeA 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(CTC)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

(28)

=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 irank(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 ATA 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 to be n n. 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 ATA.

5.2.3 Square decomposed into rectangular

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

] [ D 00 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 = BTB

where B is a unique upper triangular matrix.


6 GENERAL STATISTICS AND PROBABILITY

6 General Statistics and Probability

6.1 Moments of any distribution

6.1.1 Mean and covariance of linear forms

Assume X and x to be a matrix and a vector of random variables. Then

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

Var[Ax] = AVar[x]AT

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

See [14].

6.1.2 Mean and Variance of Square 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

E[xTAx] = Tr(A) + cTAc

Var[xTAx] = 222Tr(A2) + 42cTA2c+ 43cTAa+ (4 322)aTa

See [14]

6.2 Expectations

Assume x to be a stochastic vector with mean m, covariance M and centralmoments vr = E[(xm)r].

6.2.1 Linear Forms

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

E[x+ b] = m+ b

6.2.2 Quadratic Forms

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

E[xxT ] = M+mmT

E[xaTx] = (M+mmT )aE[xTaxT ] = aT (M+mmT )

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

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


6.2 Expectations 6 GENERAL STATISTICS AND PROBABILITY

E[(Ax+ a)T (Bx+ b)] = Tr(AMBT ) + (Am+ a)T (Bm+ b)E[xTx] = Tr(M) +mTm

E[xTAx] = Tr(AM) +mTAmE[(Ax)T (Ax)] = Tr(AMAT ) + (Am)T (Am)

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

See [7].

6.2.3 Cubic Forms

Assume x to be independent, then

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

E[xxTx] = v3 + 2Mm+ (Tr(M) +mTm)mE[(Ax+ a)(Ax+ a)T (Ax+ a)] = Adiag(ATA)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 )

See [7].


7 GAUSSIANS

7 Gaussians

7.1 One Dimensional

7.1.1 Density and Normalization

The density is

p(s) =12pi2

exp( (s )

2

22

)Normalization integrals

e(s)222 ds =

2pi2

e(ax

2+bx+c)dx =pi

aexp

[b2 4ac

4a

]ec2x

2+c1x+c0dx =

pi

c2 exp[c21 4c2c04c2

]7.1.2 Completing the Squares

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

a = c2 b = 12c1c2

w =14c21c2

+ c0

orc2x

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

=c12c2

2 =12c2

d = c0 c21

4c2

7.1.3 Moments

If the density is expressed by

p(x) =12pi2

exp[ (s )

2

22

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

then the first few basic moments are

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

(c12c2

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

[3 c212c2

]x4 = 4 + 622 + 34 =

(c12c2

)4+ 6

(c12c2

)2 (12c2

)+ 3

(12c2

)2and the central moments are


7.2 Basics 7 GAUSSIANS

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

[12c2

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

[12c2

]2A kind of pseudo-moments (un-normalized integrals) can easily be derived as

exp(c2x2 + c1x)xndx = Zxn =

pi

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

]7.2 Basics

7.2.1 Density and normalization

The density of x N (m,) is

p(x) =1

det(2pi)exp

[12(xm)T1(xm)

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

exp[12(xm)T1(xm)

]dx =

det(2pi)

exp

[12xTAx+ bTx

]dx =

det(2piA1) exp

[12bTA1b

]

exp[12Tr(STAS) + Tr(BTS)

]dS =

det(2piA1) exp

[12Tr(BTA1B)

]The derivatives of the density are

p(x)x

= p(x)1(xm)

2p

xxT= p(x)

(1(xm)(xm)T1 1

)7.2.2 Linear combination

Assume x N (mx,x) and y N (my,y) thenAx+By + c N (Amx +Bmy + c,AxAT +ByBT )


7.2 Basics 7 GAUSSIANS

7.2.3 Rearranging Means

NAx[m,] =det(2pi(AT1A)1)

det(2pi)Nx[A1m, (AT1A)1]

7.2.4 Rearranging into squared form

If A is symmetric, then

12xTAx+ bTx = 1

2(xA1b)TA(xA1b) + 1

2bTA1b

12Tr(XTAX)+Tr(BTX) = 1

2Tr[(XA1B)TA(XA1B)]+1

2Tr(BTA1B)

7.2.5 Sum of two squared forms

In vector formulation (assuming 1,2 are symmetric)

12(xm1)T11 (xm1)

12(xm2)T12 (xm2)

= 12(xmc)T1c (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

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

12Tr((XM1)T11 (XM1))

12Tr((XM2)T12 (XM2))

= 12Tr[(XMc)T1c (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)


7.3 Moments 7 GAUSSIANS

7.2.6 Product of gaussian densities

Let Nx(m,) denote a density of x, then

Nx(m1,1) Nx(m2,2) = ccNx(mc,c)

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

1det(2pi(1 +2))

exp[12(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.

7.3 Moments

7.3.1 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

7.3.2 Mean and variance of square forms

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

E(xxT ) = +mmT

E[xTAx] = Tr(A) +mTAmVar(xTAx) = 24Tr(A2) + 42mTA2m

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

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

Cov(xTAx,xTBx) = 24Tr(AB)


7.3 Moments 7 GAUSSIANS

7.3.3 Cubic forms

E[xbTxxT ] = mbT (M+mmT ) + (M+mmT )bmT

+bTm(MmmT )

7.3.4 Mean of Quartic Forms

E[xxTxxT ] = 2(+mmT )2 +mTm(mmT )+Tr()(+mmT )

E[xxTAxxT ] = (+mmT )(A+AT )(+mmT )+mTAm(mmT ) + Tr[A(+mmT )]

E[xTxxTx] = 2Tr(2) + 4mTm+ (Tr() +mTm)2

E[xTAxxTBx] = Tr[A(B+BT )] +mT (A+AT )(B+BT )m+(Tr(A) +mTAm)(Tr(B) +mTBm)

E[aTxbTxcTxdTx]= (aT (+mmT )b)(cT (+mmT )d)

+(aT (+mmT )c)(bT (+mmT )d)+(aT (+mmT )d)(bT (+mmT )c) 2aTmbTmcTmdTm

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 ]

E[(Ax+ a)T (Bx+ b)(Cx+ c)T (Dx+ d)]= Tr[A(CTD+DTC)BT ]

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

See [7].

7.3.5 Moments

E[x] =k

kmk

Cov(x) =k

k

kk(k +mkmTk mkmTk)


7.4 Miscellaneous 7 GAUSSIANS

7.4 Miscellaneous

7.4.1 Whitening

Assume x N (m,) thenz = 1/2(xm) N (0, I)

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

7.4.2 The Chi-Square connection

Assume x N (m,) and x to be n dimensional, thenz = (xm)T1(xm) 2n

7.4.3 Entropy

Entropy of a D-dimensional gaussian

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

det(2pi) D

2

7.5 One Dimensional Mixture of Gaussians

7.5.1 Density and Normalization

p(s) =Kk

k2pi2k

exp[12(s k)2

2k

]

7.5.2 Moments

An useful fact of MoG, is that

xn =k

kxnk

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

p(x) =k

k12pi2k

exp[12(x k)2

2k

]p(x) =

k

kCk exp[ck2x

2 + ck1x]

as


7.6 Mixture of Gaussians 7 GAUSSIANS

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 [( 12ck2)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 + 62k2k + 34k (2k + 2k)(2k + 2k)]7.6 Mixture of Gaussians

7.6.1 Density

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

p(x) =Kk=1

k1

det(2pik)exp

[12(xmk)T1k (xmk)

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


8 MISCELLANEOUS

8 Miscellaneous

8.1 Functions and Series

8.1.1 Finite Series

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

8.1.2 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)TH(x0)(x x0)

where

g(x0) =f(x)x

x0

H(x0) =2f(x)xxT

x0

8.1.3 Taylor Expansion of Vector Functions

8.1.4 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

8.2 Indices, Entries and Vectors 8 MISCELLANEOUS

8.1.5 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+ 1

2A2 ...

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

8.1.6 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 ...

8.2 Indices, Entries and Vectors

Let ei denote the column vector which is 1 on entry i and zero elsewhere, i.e.(ei)j = ij , and let Jij denote the matrix which is 1 on entry (i, j) and zeroelsewhere.

8.2.1 Rows and Columns

i.th row of A = eTi A

j.th column of A = Aej



8.2.2 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 =

eT2AeT1AeT3A

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.

8.2.3 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 the placed of the j.thcolumn. Assume A to be nm and Jij to be p n

JijA =

00...Aj...0

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

8.2.4 Rewriting product of elements

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

AikBlj = (ATeieTj BT )kl = (ATJijBT )kl

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

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



8.2.5 Properties of the Singleentry Matrix

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

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

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

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

8.2.6 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) = (ATBT )ij

Tr(AJjiB) = (BA)ij

Tr(AJijJijB) = diag(ATBT )ij

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

xTAJijBx = (ATxxTBT )ij

xTAJijJijBx = diag(ATxxTBT )ij

8.2.7 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 Solutions to Systems of Equations 8 MISCELLANEOUS

8.3 Solutions to Systems of Equations

8.3.1 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

8.3.2 Standard Square

Assume A is square and invertible, then

Ax = b x = A1b

8.3.3 Degenerated Square

8.3.4 Over-determined Rectangular

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

Ax = b x = (ATA)1ATb = 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].

8.3.5 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 solutionsX. ButXmin is the solution which minimizes||AXB||2 and also the solution with the smallest norm ||X||2. See [3].


8.4 Block matrices 8 MISCELLANEOUS

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.

8.3.6 Linear form and zeros

Ax = 0, x A = 0

8.3.7 Square form and zeros

If A is symmetric, then

xTAx = 0, x A = 0

8.4 Block matrices

Let Aij denote the ij.th block of A.

8.4.1 Multiplication

Assuming the dimensions of the blocks matches we have[A11 A12A21 A22

] [B11 B12B21 B22

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

]8.4.2 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)

8.4.3 The Inverse

The inverse can be expressed as by the use of

C1 = A11 A12A122 A21C2 = A22 A21A111 A12


8.5 Matrix Norms 8 MISCELLANEOUS

as [A11 A12A21 A22

]1=[

C11 A111 A12C12C12 A21A111 C12

]=[A111 +A

111 A12C

12 A21A

111 C11 A12A122

A122 A21C11 A122 +A122 A21C11 A12A122

]8.4.4 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)

8.5 Matrix Norms

8.5.1 Definitions

A matrix norm is a mapping which fulfils

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

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

8.5.2 Examples

||A||1 = maxj

i

|Aij |

||A||2 =max eig(ATA)

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

||A|| = maxi

j

|Aij |

||A||F =

ij

|Aij |2 (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.


8.6 Positive Definite and Semi-definite Matrices 8 MISCELLANEOUS

8.5.3 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 nn

n

n

||A||2mn

n

m 1 1

||A||Fmn

n

m

d 1

||A||KFmnd

nd

md d

d

which are to be read as, e.g.

||A||2 m ||A||

8.6 Positive Definite and Semi-definite Matrices

8.6.1 Definitions

A matrix A is positive definite if and only if

xTAx > 0, x

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

xTAx 0, x

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

8.6.2 Eigenvalues

The following holds with respect to the eigenvalues:

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

8.6.3 Trace

The following holds with respect to the trace:

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

8.6.4 Inverse

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


8.7 Integral Involving Dirac Delta Functions 8 MISCELLANEOUS

8.6.5 Diagonal

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

8.6.6 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

8.6.7 Decomposition II

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

8.6.8 Equation with zeros

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

8.6.9 Rank of product

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

8.6.10 Positive definite property

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

8.6.11 Outer Product

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

8.6.12 Small pertubations

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

8.7 Integral Involving Dirac Delta Functions

Assuming A to be square, thenp(s)(xAs)ds = 1

det(A)p(A1x)

Assuming A to be underdetermined, i.e. tall, thenp(s)(xAs)ds =

{1

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

0 elsewhere

}


8.8 Miscellaneous 8 MISCELLANEOUS

See [8].

8.8 Miscellaneous

For any A it holds that

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

Assume A is positive definite. Then

rank(BTAB) = rank(B)

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


A PROOFS AND DETAILS

A Proofs and Details

A.0.1 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. 8.2.5, the resultfollows easily.

A.0.2 Details on Eq. 47

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)]

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

det(XHAX)

A PROOFS AND DETAILS

Hence, derivative yields

det(XHAX)X

=12

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

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

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

[12] Laurent Schwartz. Cours dAnalyse, volume II. Hermann, Paris, 1967. Asreferenced in [9].

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

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

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

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

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


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