The Kronecker Product A Product of the Times

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1 The Kronecker Product A Product of the Times Charles Van Loan Department of Computer Science Cornell University

2 The Kronecker Product B C is a block matrix whose ij-th block is b ij C. E.g., [ b11 b 12 b 21 b 22 ] C = b 11C b 12 C b 21 C b 22 C Also called the Direct Product or the Tensor Product

3 Every b ij c kl Shows Up [ b11 b 12 b 21 b 22 ] c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 = b 11 c 11 b 11 c 12 b 11 c 13 b 12 c 11 b 12 c 12 b 12 c 13 b 11 c 21 b 11 c 22 b 11 c 23 b 12 c 21 b 12 c 22 b 12 c 23 b 11 c 31 b 11 c 32 b 11 c 33 b 12 c 31 b 12 c 32 b 12 c 33 b 21 c 11 b 21 c 12 b 21 c 13 b 22 c 11 b 22 c 12 b 22 c 13 b 21 c 21 b 21 c 22 b 21 c 23 b 22 c 21 b 22 c 22 b 22 c 23 b 21 c 31 b 21 c 32 b 21 c 33 b 22 c 31 b 22 c 32 b 22 c 33

4 Basic Algebraic Properties (B C) T = B T C T (B C) 1 = B 1 C 1 (B C)(D F) = BD CF B (C D) C B = (B C) D = (Perfect Shuffle) T (B C)(Perfect Shuffle)

5 Reshaping KP Computations Suppose B,C IR n n and x IR n2. The operation y = (B C)x is O(n 4 ): y = kron(b,c)*x The operation Y = CXB T is O(n 3 ): y = reshape(c*reshape(x,n,n)*b,n,n)

6 Talk Outline 1. The 1800 s Origins: Z 2. The 1900 s Heightened Profile: 3. The 2000 s Future:

7 The 1800 s Z

8 Products and Deltas δ ij Leopold Kronecker ( )

9 The Kronecker Delta: Not as Interesting! U T δ ij V = δij κ 2 (δ ij ) = 1 δ ij

10 Acknowledgement H.V. Henderson, F. Pukelsheim, and S.R. Searle (1983). On the History of the Kronecker Product, Linear and Multilinear Algebra 14, Shayle Searle, Professor Emeritus, Cornell University (right)

11 Scandal! H.V. Henderson, F. Pukelsheim, and S.R. Searle (1983). On the History of the Kronecker Product, Linear and Multilinear Algebra 14, Abstract History reveals that what is today called the Kronecker 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 A.N. Langville and W.J. Stewart (2004). The Kronecker product and stochastic automata networks, J. Computational and Applied Mathematics 167,

12 Let s Get to the Bottom of This! History Reveals That What is Today Called The Kronecker Product Should Be Called The Zehfuss Product

13 Who Was Zeyfuss? Born Obscure professor of mathematics at University of Heidelberg for a while. Then went on to other things. Wrote papers on determinants... G. Zeyfuss (1858). Uber eine gewasse Determinante, Zeitscrift fur Mathematik und Physik 3,

14 Main Result a.k.a. The Z Theorem If B IR m m and C IR n n then det (B C) = det(b) n det(c) m Proof: Noting that I n B and I m C are block diagonal, take determinants in B C = (B I n )(I m C) = P(I n B)P T (I m C) where P is a perfect shuffle permutation.

15 Zehfuss(1858) = a 1 A 1 a 1 B 1 b 1 A 1 b 1 B 1 c 1 A 1 c 1 B 1 d 1 A 1 d 1 B 1 a 1 A 2 a 1 B 2 b 1 A 2 b 1 B 2 c 1 A 2 c 1 B 2 d 1 A 2 d 1 B 2 a 2 A 1 a 2 B 1 b 2 A 1 b 2 B 1 c 2 A 1 c 2 B 1 d 2 A 1 d 2 B 1 a 2 A 2 a 2 B 2 b 2 A 2 b 2 B 2 c 2 A 2 c 2 B 2 d 2 A 2 d 2 B 2 a 3 A 1 a 3 B 1 b 3 A 1 b 3 B 1 c 3 A 1 c 3 B 1 d 3 A 1 d 3 B 1 a 3 A 2 a 3 B 2 b 3 A 2 b 3 B 2 c 3 A 2 c 3 B 2 d 3 A 2 d 3 B 2 a 4 A 1 a 4 B 1 b 4 A 1 b 4 B 1 c 4 A 1 c 4 B 1 d 4 A 1 d 4 B 1 a 4 A 2 a 4 B 2 b 4 A 2 b 4 B 2 c 4 A 2 c 4 B 2 d 4 A 2 d 4 B 2

16 Zehfuss(1858) p = a 1 b 1 c 1 d 1 a 2 b 2 c 2 d 2 a 3 b 3 c 3 d 3 a 4 b 4 c 4 d 4 und P = A 1 B 1 A 2 B 2 2 2,2 = p 2 4 P Mm = p M P m

17 Hensel (1891) Student in Berlin Maintains that Kronecker presented the Z-theorem in his lectures. K. Hensel (1891). Uber die Darstellung der Determinante eines Systems, welsches aus zwei a irt ist, ACTA Mathematica 14,

18 The 1900 s

19 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.

20 Rutherford(1933) On the Condition that Two Zehfuss Matrices are Equal B C??? = F G If B (m b n b ), C (m c n c ), F (m f n f ), G (m g n g ), then (B C) ij = (F G) ij means B(floor(i/m c ), floor(j/n c )) C(i mod m c,j mod n c ) = F(floor(i/m g ), floor(j/n g )) G(i mod m g,j mod n g )

21 Van Loan(1934) On the Condition that Two Square Zehfuss Matrices are Equal B C??? = F G If then U T b BV b = diag(σ i (B)) U T c CV c = diag(σ i (C)) U T b FV b = α diag(σ i (F)) U T c GV c = 1 α diag(σ i(g))

22 Heightened Profile Beginning in the 60s Some Reasons: Regular Grids Tensoring Low Dimension Ideas Higher Order Statistics Fast Transforms Preconditioners Numerical Multi-linear Algebra

23 Regular Grids (M +1)-by-(N +1) discretization of the Laplacian on a rectangle... A = I M T N + T M I N T 5 =

24 Tensoring Low Dimension Ideas b a f(x) dx n i=1 w i f(x i ) = w T f(x) b1 b2 b3 a 1 a 2 a 3 f(x,y,z) dxdy dz n x i=1 n y j=1 n z k=1 w (x) i w (y) j w (z) k f(x i,y j,z k ) = (w (x) w (y) w (z) ) T f(x y z)

25 Higher Order Statistics E(xx T ) E(x x) E(x x x) Kronecker powers: k A = A A A (k times)

26 Fast Transforms FFT B 4 = F 16 P 16 = B 16 (I 2 B 8 )(I 4 B 4 )(I 8 B 2 ) ω ω 4 ω n = exp( 2πi/n) Haar Wavelet Transform W 2m = [ W m ( 1 1 ) I m ( 1 1 ) ] if m > 1 [ 1 ] if m = 1.

27 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. Best Example: A band block Toeplitz with banded Toeplitz blocks. B and C chosen to be band Toeplitz. Nagy, Kamm, Kilmer, Perrone, others

28 Tensor Decompositions/Approximation Given A = A(1:n, 1:n, 1:n, 1:n), find orthogonal Q = [ q 1 q n ] U = [ u 1 u n ] V = [ v 1 v n ] W = [ w 1 w n ] and a core tensor σ so n vec(a) σ ijk,l w i v j u k q l i,j,k,l=1

29 Offspring 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 Symmetric Kronecker Product 7. The Bi-Alternate Product

30 Left Kronecker Product Definition: ] [ c11 B c 12 B ] B Left [ c11 c 12 c 21 c 22 = c 21 B c 22 B = C B Fact: If B IR m b n b and C IR m c n c then B Left C = Π T m c, m b m c (B C)Π nc, n b n c Perfect Shuffles

31 The Hadamard Product Definition: b 11 b 12 b 21 b 22 Had c 11 c 12 c 21 c 22 = b 11 c 11 b 12 c 12 b 21 c 21 b 22 c 22 b 31 b 32 c 31 c 32 b 31 c 31 b 32 c 32 B Had C = B. C

32 The Hadamard Product Fact: If à = B C and B,C IR m n, then B Had C = Ã(1:(m+1):m2, 1:(n+1):n 2 ) E.g., b 11 b 12 b 21 b 22 b 31 b 32 c 11 c 12 c 21 c 22 c 31 c 32 = b 11 c 11 b 11 c 12 b 12 c 11 b 12 c 12 b 11 c 21 b 11 c 22 b 12 c 21 b 12 c 22 b 11 c 31 b 11 c 32 b 12 c 31 b 12 c 32 b 21 c 11 b 21 c 12 b 22 c 11 b 22 c 12 b 21 c 21 b 21 c 22 b 22 c 21 b 22 c 22 b 21 c 31 b 21 c 32 b 22 c 31 b 22 c 32 b 31 c 11 b 31 c 12 b 32 c 11 b 32 c 12 b 31 c 21 b 31 c 22 b 32 c 21 b 32 c 22 b 31 c 31 b 31 c 32 b 32 c 31 b 32 c 32

33 The Tracy-Singh Product Definition: B = B TS C = B 11 B 12 B 21 B 22 B 31 B 32 C = [ C11 C 12 C 21 C 22 B 11 C 11 B 11 C 12 B 12 C 11 B 12 C 12 B 11 C 21 B 11 C 22 B 12 C 21 B 12 C 22 B 21 C 11 B 21 C 12 B 22 C 11 B 22 C 12 B 21 C 21 B 21 C 22 B 22 C 21 B 22 C 22 B 31 C 11 B 31 C 12 B 32 C 11 B 32 C 12 B 31 C 21 B 31 C 22 B 32 C 21 B 32 C 22 ]

34 The Khatri-Rao Product Definition: B = B 11 B 12 B 21 B 22 C = C 11 C 12 C 21 C 22 B 31 B 32 B K-R C = C 31 C 32 B 11 C 11 B 12 C 12 B 21 C 21 B 22 C 22 B 31 C 31 B 32 C 32

35 The Khatri-Rao Product Fact: B TS C = B KR C is a submatrix of B TS C B 11 C 11 B 11 C 12 B 12 C 11 B 12 C 12 B 11 C 21 B 11 C 22 B 12 C 21 B 12 C 22 B 11 C 31 B 11 C 32 B 12 C 31 B 12 C 32 B 21 C 11 B 21 C 12 B 22 C 11 B 22 C 12 B 21 C 21 B 21 C 22 B 22 C 21 B 22 C 22 B 21 C 31 B 21 C 32 B 22 C 31 B 22 C 32 B 31 C 11 B 31 C 12 B 32 C 11 B 32 C 12 B 31 C 21 B 31 C 22 B 32 C 21 B 32 C 22 B 31 C 31 B 31 C 32 B 32 C 31 B 32 C 32

36 The Generalized Kronecker Product Regalia and Mitra (1989), Kronecker Products, Unitary Matrices, and Signal Processing Applications, SIAM Review, 31, B 1 B 2 B 3 B 4 gen C = B 1 Left C(1, :) B 2 Left C(2, :) B 3 Left C(3, :) B 4 Left C(4, :)]

37 The Generalized Kronecker Product B 1 B 2 B 3 B 4 GEN { C1 C 2 } = { B1 B 2 { B3 B 4 } gen C 1 } gen C 2

38 The Symmetric Kronecker Product Kronecker Product turns matrix equations into vector equations: CXB T = G (B C) vec(x) = vec(g) The symmetric Kronecker product does the same thing for matrix equations with symmetric solutions: 1 2 ( CXB T + BXC T) = G (symmetric) (B sym C) svec(x) = svec(g) where svec x 11 x 12 x 13 x 12 x 22 x 23 x 13 x 23 x 33 = [ ] T x 11 2x12 x 22 2x13 2x23 x 33

39 Symmetric Kronecker Product Fact: If P = α α α α α α α α = 1/ 2 then vec(x) = P svec(x) and B sym C = P T (B C)P

40 Bi-Alternate Product B C = 1 2 (B C + C B) W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM.

41 The 2000 s Three Predictions

42 Big N Will Mean Big d Will Mean KP A = [ a11 a 12 a 21 a 22 ] [ b11 b 12 b 21 b 22 ] [ z11 z 12 z 21 z 22 ] N = 2 d

43 Think Scalar Think Block Think Tensor If for all 1 m i n we have, n i 1,i 2,i 3,i 4 =1 B(m 1,m 2,m 3,m 4 ) = W(i 1,m 1 )Y (i 2,m 2 )X(i 3,m 3 )Z(i 4,m 4 )A(i 1,i 2,i 3,i 4 ) then B = (W Y ) T A(X Z)

44 Data-Sparse Approximate Factorizations A (B 1 C 1 )(B 2 C 2 )(B 3 C 3 )

45 Det(Log(A)) A (B C)(D E)(F G) log(det(a)) n c log(det(b)) + n b log(det(c)) + n e log(det(d)) + n d log(det(e)) + n g log(det(f)) + n f log(det(g)) See Zehfuss (2058).

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