The Kronecker Product

A Product of the Times

Charles Van Loan

Department of Computer Science

Cornell University

Presented at the SIAM Conference on Applied Linear Algebra, Monterey, Califirnia,

October 26, 2009

The Kronecker Product

B ⊗ C is a block matrix whose ij-th block is bijC.

E.g.,

[

b11 b12b21 b22

]

⊗ C =

b11C b12C

b21C b22C

Also called the “Direct Product” or the “Tensor Product”

Every bijckl Shows Up

[

b11 b12b21 b22

]

⊗

c11 c12 c13c21 c22 c23c31 c32 c33

=

b11c11 b11c12 b11c13 b12c11 b12c12 b12c13

b11c21 b11c22 b11c23 b12c21 b12c22 b12c23

b11c31 b11c32 b11c33 b12c31 b12c32 b12c33

b21c11 b21c12 b21c13 b22c11 b22c12 b22c13

b21c21 b21c22 b21c23 b22c21 b22c22 b22c23

b21c31 b21c32 b21c33 b22c31 b22c32 b22c33

Basic Algebraic Properties

(B ⊗ C)T = BT ⊗ CT

(B ⊗ C)−1 = B−1 ⊗ C−1

(B ⊗ C)(D ⊗ F ) = BD ⊗ CF

B ⊗ (C ⊗ D) = (B ⊗ C) ⊗ D

C ⊗ B = (Perfect Shuffle)T (B ⊗ C)(Perfect Shuffle)

R.J. Horn and C.R. Johnson(1991). Topics in Matrix Analysis, Cambridge University Press, NY.

Reshaping KP Computations

Suppose B,C ∈ IRn×n and x ∈ IRn2.

The operation y = (B ⊗ C)x is O(n4):

y = kron(B,C)*x

The equivalent, reshaped operation Y = CXBT is O(n3):

y = reshape(C*reshape(x,n,n)*B’,n,n)

H.V. Henderson and S.R.Searle (1981). “The Vec-Permutation Matrix, The Vec Operator,and Kronecker Products, A Review,” Linear and Mulitilinear Algebra, 9, 271–288.

Talk Outline

1. The 1800’s

Origins: ©Z2. The 1900’s

Heightened Profile: ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗3. The 2000’s

Future: ©∞

The 1800’s

©Z

Products and Deltas

⊗ δij

Leopold Kronecker (1823–1891)

Of course, the contributions go far beyond this...

E.T. Bell (1937). Men of Mathematics, Simon and Schuster, New York.

K. Hensel (1968). Leopold Kronecker’s Werke, Chelsea Publishing Company, New York.

Brief Survey of the Kronecker Delta

UT δijV =∣

∣δij

∣

∣

κ2(δij) = 1δij

G.H. Golub and C. Van Loan (1996). Matrix Computations, 3rd Ed., Johns Hopkins University Press,Baltimore, Maryland.

Acknowledgement

H.V. Henderson, F. Pukelsheim, and S.R. Searle (1983). “Onthe History of the Kronecker Product,” Linear and Multilinear

Algebra 14, 113–120.

Shayle Searle, Professor Emeritus, Cornell University (right)

Scandal!

H.V. Henderson, F. Pukelsheim, and S.R. Searle (1983). “Onthe History of the Kronecker Product,” Linear and Multilinear

Algebra 14, 113–120.

Abstract

History reveals that what is today called the Kro-necker product should be called the Zehfuss Product.

This fact is somewhat appreciated by the modern (numerical) linear algebra community:

R.J. Horn and C.R. Johnson(1991). Topics in Matrix Analysis, Cambridge University Press, NY, p. 254.

A.N. Langville and W.J. Stewart (2004). “The Kronecker product and stochastic automata networks,” J.

Computational and Applied Mathematics 167, 429–447.

Who Was Zehfuss?

Born 1832.

Obscure professor of mathematics at University of Heidelberg fora while. Then went on to other things.

Wrote papers on determinants...

G. Zehfuss (1858). “Uber eine gewisse Determinante,” Zeitschrift fur Mathematik undPhysik 3, 298–301.

Main Result a.k.a. “The Z Theorem”

If B ∈ IRm×m and C ∈ IRn×n then

det (B ⊗ C) = det(B)ndet(C)m

Modern Proof

Note that In ⊗ B and Im ⊗ C are block diagonal and take determinants in

B ⊗ C = (B ⊗ In)(Im ⊗ C) = P (In ⊗ B)P T (Im ⊗ C)

where P is a perfect shuffle permutation.

Excerpts from Zehfuss(1858)

∆ =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

a1A1 a1B1 b1A1 b1B1 c1A1 c1B1 d1A1 d1B1

a1A2 a1B2 b1A2 b1B2 c1A2 c1B2 d1A2 d1B2

a2A1 a2B1 b2A1 b2B1 c2A1 c2B1 d2A1 d2B1

a2A2 a2B2 b2A2 b2B2 c2A2 c2B2 d2A2 d2B2

a3A1 a3B1 b3A1 b3B1 c3A1 c3B1 d3A1 d3B1

a3A2 a3B2 b3A2 b3B2 c3A2 c3B2 d3A2 d3B2

a4A1 a4B1 b4A1 b4B1 c4A1 c4B1 d4A1 d4B1

a4A2 a4B2 b4A2 b4B2 c4A2 c4B2 d4A2 d4B2

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

Excerpts from Zehfuss(1858)

p =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

a1 b1 c1 d1

a2 b2 c2 d2

a3 b3 c3 d3

a4 b4 c4 d4

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

und P =

∣

∣

∣

∣

∣

A1 B1

A2 B2

∣

∣

∣

∣

∣

∆2 2,2 = p24P

42

∆2 Mm = pMPm

Hensel (1891)

Student in Berlin 1880-1884.

Maintains that Kronecker presented the Z-theorem in his lectures.

K. Hensel (1891). “Uber die Darstellung der Determinante eines Systems, welches aus zwei anderencomponirt ist,” ACTA Mathematica 14, 317–319.

The 1900’s

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

Muir (1911)

Attributes the Z-theorem to Zehfuss.

Calls det(B ⊗ C) the “Zehfuss determinant.”

T. Muir (1911). The Theory of Determinants in the Historical Order of Development, Vols 1-4, Dover,NY.

Rutherford(1933)

Q. When are two Zehfuss matrices equal?

B ⊗ C???= F ⊗ G

Subscripting from zero, if B (mb×nb), C (mc×nc), F (mf×nf ),G (mg × ng), then (B ⊗ C)ij = (F ⊗ G)ij means

B(floor(i/mc), floor(j/nc)) · C(i mod mc, j mod nc)

=F (floor(i/mg), floor(j/ng)) · G(i mod mg, j mod ng)

D.E. Rutherford (1933). “On the Condition that Two Zehfuss Matrices are Equal,” Bull. Amer. Math.

Soc. 39, 801-808.

©Z → ⊗ Why?

“...a series of influential texts at and after the turn of thecentury permanently associated Kronecker’s name with the“ ⊗ ” product and this terminology is nearly universal to-day.”

Horn and Johnson (1991)

“...the textbook of Scott and Matthews (1904) which ap-peared four years after the publication of Rados’ paper,gave new life to the old error. This was probably due tothe teaching of Pascal, whose second edition (1923) stillpropagates the error [of the first edition (1897).]”

Muir (1927)

Heightened Profile Beginning in the 60s

Some Reasons

Regular Grids

Tensoring Low Dimension Ideas

Higher Order Statistics

Fast Transforms

Preconditioners

Quantum Computing

Tensor Decompositions/Approximations

C. Van Loan (2000). “The Ubiquitous Kronecker Product,” Journal of Computational and Applied Math-

ematics,, 85-100.

Regular Grids

(M +1)-by-(N +1) discretization of the Laplacian on a rectangle...

A = IM ⊗ TN + TM ⊗ IN

T5 =

2 −1 0 0 0−1 2 −1 0 0

0 −1 2 −1 00 0 −1 2 −10 0 0 −1 2

F.W. Dorr (1970). “The Direct Solution of the Discrete Poisson Equation on a Rectangle,” SIAM Review

12, 248–263.

G.H. Golub and C.F. Van Loan (1996). Matrix Computations, 3rd Ed, Johns Hopkins University Press,

Baltimore, MD.

Tensoring Low Dimension Ideas

∫ b

af (x) dx ≈

n∑

i=1

wi f (xi) = wTf (x)

∫ b1

a1

∫ b2

a2

∫ b3

a3

f (x, y, z) dx dy dz ≈nx∑

i=1

ny∑

j=1

nz∑

k=1

w(x)i w

(y)j w

(z)k f (xi, yj, zk)

= (w(x) ⊗ w(y) ⊗ w(z))Tf (x ⊗ y ⊗ z)

A. Graham (1981). Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Ltd, Chich-ester, England.

Higher Order Statistics

E(xxT )⇓

E(x ⊗ x)⇓

E(x ⊗ x ⊗ · · · ⊗ x)

Kronecker powers:

⊗ kA = A ⊗ A ⊗ · · · ⊗ A (k times)

T.F. Andre, R.D. Nowak, and B.D. Van Veen (1997). “Low Rank Estimation of Higher Order Statistics,”IEEE Trans. Signal Processing 45, 673–685.

Fast Transforms

FFT

F16P16 = B16(I2 ⊗ B8)(I4 ⊗ B4)(I8 ⊗ B2)

B4 =

1 0 1 00 1 0 ω41 0 −1 00 1 0 −ω4

ωn = exp(−2πi/n)

J. Granata, M. Conner, and R. Tolimieri (1992). “Recursive Fast Algorithms and the Role of the Tensor

Product,” IEEE Transactions on Signal Processing 40, 2921–2930.

C. Van Loan(1992). Computational Frameworks for the Fast Fourier Transform, SIAM Publications,

Philadelphia, PA.

Fast Transforms Cont’d

Haar Wavelet Transform

W2m =

[

Wm ⊗(

11

)

Im ⊗(

1−1

) ]

if m > 1

[ 1 ] if m = 1

.

Fast Gauss Transform

gij = exp(−‖ sj − ti ‖22/δ) ⇒ G = Gnear + Gfar

Gnear involves a Kronecker Product

G. Strang(1993). “Wavelet Transforms Versus Fourier Transforms,” Bulletin of the American Mathematical

Association, 28, 288–305.

X. Sun and Y. Bao (2003). “A Kronecker Product Represenation of the Fast Gauss Transform,” SIAM J.

Matrix Anal. Appl., 24, 768–786.

Preconditioners

If A ≈ B ⊗ C, then B ⊗ C has potential as a preconditioner.

It captures the essence of A.

It is easy to solve (B ⊗ C)z = r.

Good Example: A band block Toeplitz with banded Toeplitzblocks. B and C chosen to be band Toeplitz.

J. Kamm and J.G. Nagy (2000). “Optimal Kronecker Product Approximation of Block Toeplitz Matrices,”

SIAM J. Matrix Anal. and Appl., 22, 155–172.

J. Nagy and M. Kilmer (2006). “Kronecker Product Approximation for Three-Dimensional Imaging Ap-plications,” IEEE Trans. Image Proc. 15, 604-613.

Quantum Computing

Filled with Kronecker powers of 2-by-2’s.

H ⊗ n = H ⊗ H ⊗ · · · ⊗ H H =1√2

[

1 1−1 1

]

N.D. Mermin (2007). Quantum Computer Science, Cambridge University Press, Cambridge, England.

Tensor Decompositions/Approximation

E.g. Given A = A(1:n, 1:n, 1:n, 1:n), find orthogonal

Q =[

q1 · · · qn]

U =[

u1 · · · un]

V =[

v1 · · · vn]

W =[

w1 · · · wn]

and a “core tensor” σ so

vec(A) ≈n

∑

i,j,k,`=1

σijk,` wi ⊗ vj ⊗ uk ⊗ q`

T. Kolda and B. Bader (2009). “Tensor Decompositions and Applications,” SIAM Review 51, 455–500.

Descendants

1. The Left Kronecker Product

2. The Hadamard Product

3. The Tracy-Singh Product

4. The Khatri-Rao Product

5. The Generalized Kronecker Product

6. The Strong Kronecker Product

7. The Symmetric Kronecker Product

8. The Bi-Alternate Product

Left Kronecker Product

Definition:

BLeft⊗

[

c11 c12

c21 c22

]

=

[

c11B c12B

c21B c22B

]

= C ⊗ B

Fact:

If B ∈ IRmb×nb and C ∈ IRmc×nc then

BLeft⊗ C = ΠT

mc, mbmc(B ⊗ C)Πnc, nbnc

↑ Perfect Shuffles ↑

F.A. Graybill(1969). Introduction to Matrices with Applications in Statistics, Wadsworth, Belmont, CA.

The Hadamard Product

Definition:

b11 b12

b21 b22

b31 b32

Had⊗

c11 c12

c21 c22

c31 c32

=

b11c11 b12c12

b21c21 b22c22

b31c31 b32c32

BHad⊗ C = B. ∗ C

A. Smilde, R. Bro, and P. Geladi (2004). Multiway Analysis, John Wiley, Chichester, England.

The Hadamard Product

Fact:

If A = B ⊗ C and B,C ∈ IRm×n , then

BHad⊗ C = A(1:(m+1):m2, 1:(n+1):n2)

E.g.,

b11 b12

b21 b22

b31 b32

⊗

c11 c12

c21 c22

c31 c32

=

b11c11 b11c12 b12c11 b12c12

b11c21 b11c22 b12c21 b12c22

b11c31 b11c32 b12c31 b12c32

b21c11 b21c12 b22c11 b22c12

b21c21 b21c22 b22c21 b22c22

b21c31 b21c32 b22c31 b22c32

b31c11 b31c12 b32c11 b32c12

b31c21 b31c22 b32c21 b32c22

b31c31 b31c32 b32c31 b32c32

The Tracy-Singh Product

Definition:

B =

B11 B12

B21 B22

B31 B32

C =

[

C11 C12

C21 C22

]

BTS⊗ C =

B11 ⊗ C11 B11 ⊗ C12 B12 ⊗ C11 B12 ⊗ C12

B11 ⊗ C21 B11 ⊗ C22 B12 ⊗ C21 B12 ⊗ C22

B21 ⊗ C11 B21 ⊗ C12 B22 ⊗ C11 B22 ⊗ C12

B21 ⊗ C21 B21 ⊗ C22 B22 ⊗ C21 B22 ⊗ C22

B31 ⊗ C11 B31 ⊗ C12 B32 ⊗ C11 B32 ⊗ C12

B31 ⊗ C21 B31 ⊗ C22 B32 ⊗ C21 B32 ⊗ C22

D.S. Tracy and R.P. Singh (1972). “A New Matrix Product and Its Applications in Partitioned Matrices,”

Statistica Neerlandica 26, 143–157.

The Khatri-Rao Product

Definition:

B =

B11 B12

B21 B22

B31 B32

C =

C11 C12

C21 C22

C31 C32

BK-R⊗ C =

B11 ⊗ C11 B12 ⊗ C12

B21 ⊗ C21 B22 ⊗ C22

B31 ⊗ C31 B32 ⊗ C32

C.R. Rao and S.K. Mitra (1971). Generalized Inverse of Matrices and Applications, John Wiley and Sons,New York.

A. Smilde, R. Bro, and P. Geladi (2004). Multiway Analysis, John Wiley, Chichester, England.

The Khatri-Rao Product

Fact:

BKR⊗ C is a submatrix of B

TS⊗ C

BTS⊗ C =

B11 ⊗ C11 B11 ⊗ C12 B12 ⊗ C11 B12 ⊗ C12

B11 ⊗ C21 B11 ⊗ C22 B12 ⊗ C21 B12 ⊗ C22

B11 ⊗ C31 B11 ⊗ C32 B12 ⊗ C31 B12 ⊗ C32

B21 ⊗ C11 B21 ⊗ C12 B22 ⊗ C11 B22 ⊗ C12

B21 ⊗ C21 B21 ⊗ C22 B22 ⊗ C21 B22 ⊗ C22

B21 ⊗ C31 B21 ⊗ C32 B22 ⊗ C31 B22 ⊗ C32

B31 ⊗ C11 B31 ⊗ C12 B32 ⊗ C11 B32 ⊗ C12

B31 ⊗ C21 B31 ⊗ C22 B32 ⊗ C21 B32 ⊗ C22

B31 ⊗ C31 B31 ⊗ C32 B32 ⊗ C31 B32 ⊗ C32

The Generalized Kronecker Product

B1

B2

B3

B4

gen⊗ C =

B1

Left⊗ C(1, :)

B2

Left⊗ C(2, :)

B3

Left⊗ C(3, :)

B4

Left⊗ C(4, :)]

Regalia and Mitra (1989), “Kronecker Products, Unitary Matrices, and Signal Processing Applications,”SIAM Review, 31, 586–613.

The Generalized Kronecker Product

B1B2B3B4

GEN⊗{

C1C2

}

=

{

B1B2

}

gen⊗ C1

{

B3B4

}

gen⊗ C2

The Strong Kronecker Product

A block matrix multiplication, but with Kronecker Products in-stead matrix-matrix products, e.g.,

B11 B12

B21 B22

Strong⊗

C11 C12

C21 C22

=

B11 ⊗ C11 + B12 ⊗ C21 B11 ⊗ C12 + B12 ⊗ C22

B21 ⊗ C11 + B22 ⊗ C21 B21 ⊗ C12 + B22 ⊗ C22

W. De Launey and J. Seberry (1994), “The Strong Kronecker Product,” Journal of Combinatorial Theory,

Series A 66, 192–213.

The Symmetric Kronecker Product

The KP turns matrix equations into vector equations:

CXBT = G ⇔ (B ⊗ C) vec(X) = vec(G)

The symmetric Kronecker product does the same thing for matrixequations with symmetric solutions:

12

(

CXBT + BXCT)

= G (symmetric)

⇔

(Bsym⊗ C) svec(X) = svec(G)

where

svec

x11 x12 x13

x12 x22 x23

x13 x23 x33

=[

x11

√2x12 x22

√2x13

√2x23 x33

]T

F. Alizadeh, J-P.A. Haeberly, and M.L. Overton (1998). “Primal-Dual Interior Point Methods for Semidef-

inite Programming: Convergence Rates, Stability, and Numerical Results,” SIAM J. Optimization 8, 746–768.

Symmetric Kronecker Product

Fact: If

P =

1 0 0 0 0 00 α 0 0 0 00 0 0 α 0 00 α 0 0 0 00 0 1 0 0 00 0 0 0 α 00 0 0 α 0 00 0 0 0 α 00 0 α 0 0 1

α = 1/√

2

then vec(X) = P · svec(X) and

Bsym⊗ C = P T (B ⊗ C)P

H.V. Henderson and S.R. Searle(1998). “Vec and Vech Operators for Matrices, with Some uses in Jacobians

and Multivariate Statistics,” The Canadian Journal of Statistics 7, 65–81.

Bi-Alternate Product

BBi-Alt⊗ C =

1

2(B ⊗ C + C ⊗ B)

W. Govaerts (2000). Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM Publications,Philadelhia, PA.

The 2000’sThree Predictions

©∞

Big N Will Mean Big d Will Mean KP

A =

[

a11 a12a21 a22

]

⊗[

b11 b12b21 b22

]

⊗ · · · ⊗[

z11 z12z21 z22

]

N = 2d

G. Beylkin and M.J. Mohlenkamp (2005). “Algorithms for Numerical Analysis in High Dimensions,” SIAM

J. Scientific Computation 26, 2133–2159.

Inevitable: Scalar → Block → Tensor

Tensor-level thinking will require an ability to spot KP’s. E.g., iffor all 1 ≤ mi ≤ n we have,

B(m1,m2,m3,m4)

=n

∑

i1,i2,i3,i4=1

W (i1,m1)Y (i2,m2)X(i3,m3)Z(i4,m4)A(i1, i2, i3, i4)

then

B = (W ⊗ Y )TA(X ⊗ Z)

NSF Workshop on Future Directions in Tensor-Based Computation and Modeling, 2009.

http://www.cs.cornell.edu/cv/TenWork/FinalReport.pdf.

Data-Sparse Approximate Factorizations

New KP-based factorizations will widen the set of solvable hugeproblems.

Sample factorization...

A ≈ (B1 ⊗ C1)(B2 ⊗ C2)(B3 ⊗ C3) · · ·

Det(Log(A)) via Zehfuss

If

A ≈ (B ⊗ C)(D ⊗ E)(F ⊗ G) · · ·

then the big log det problem becomes a bunch of smaller ones...

log(det(A)) ≈nc log(det(B)) + nb log(det(C)) +

ne log(det(D)) + nd log(det(E)) +

ng log(det(F )) + nf log(det(G)) · · ·

R.P. Barry and R.K. Pace (1999). “Monte Carlo Estimates of the Log Determinant of Large Sparse

Matrices,” Linear Algebra and Its Applications 289, 41–54.