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The Kronecker Product a Product of the Times 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 b C. ⊗ ij E.g., b b b11C b12C 11 12 C = b b ⊗ 21 22 b21C b22C Also called the “Direct Product” or the “Tensor Product” Every bijckl Shows Up c11 c12 c13 b11 b12 c21 c22 c23 b21 b22 ⊗ c31 c32 c33 = b11c11 b11c12 b11c13 b12c11 b12c12 b12c13 b11c21 b11c22 b11c23 b12c21 b12c22 b12c23 b c b c b c b c b c b c 11 31 11 32 11 33 12 31 12 32 12 33 b c b c b c b c b c b c 21 11 21 12 21 13 22 11 22 12 22 13 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 2 Suppose B, C IRn n and x IRn . ∈ × ∈ 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: Z 2. The 1900’s Heightened Profile: ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 3. The 2000’s Future: ∞ The 1800’s Z Products and Deltas δ ⊗ ij 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 T U δijV = δij 1 κ2(δ ) = ij δij 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). “On the 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). “On the 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 for a while. Then went on to other things. Wrote papers on determinants... G. Zehfuss (1858). “Uber¨ eine gewisse Determinante,” Zeitschrift f¨ur Mathematik und Physik 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 ⊗ ⊗ T B C = (B In)(Im C) = P (In B)P (Im C) ⊗ ⊗ ⊗ ⊗ ⊗ where P is a perfect shuffle permutation. Excerpts from Zehfuss(1858) a a b b c c d d 1A1 1B1 1A1 1B1 1A1 1B1 1A1 1B1 a1 2 a1 2 b1 2 b1 2 c1 2 c1 2 d1 2 d1 2 A B A B A B A B a2 1 a2 1 b2 1 b2 1 c2 1 c2 1 d2 1 d2 1 A B A B A B A B a2 2 a2 2 b2 2 b2 2 c2 2 c2 2 d2 2 d2 2 A B A B A B A B ∆ = a a b b c c d d 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 A B A B A B A B a3 2 a3 2 b3 2 b3 2 c3 2 c3 2 d3 2 d3 2 A B A B A B A B a4 1 a4 1 b4 1 b4 1 c4 1 c4 1 d4 1 d4 1 A B A B A B A B a a b b c c d d 4A2 4B2 4A2 4B2 4A2 4B2 4A2 4B2 Excerpts from Zehfuss(1858) a1 b1 c1 d1 a2 b2 c2 d2 1 1 p = und P = A B a3 b3 c3 d3 2 2 A B a b c d 4 4 4 4 2 4 ∆2 2,2 = p4P2 M m ∆2 Mm = p P 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 anderen componirt 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 (m n ), C (mc nc), F (m n ), b× b × f × f G (mg ng), then (B C) = (F G) means × ⊗ ij ⊗ ij 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 the century 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 to the teaching of Pascal, whose second edition (1923) still propagates 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 = I T + T I M ⊗ N M ⊗ N 2 1 0 0 0 1− 2 1 0 0 − − T = 0 1 2 1 0 5 − − 0 0 1 2 1 − − 0 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 n T f(x) dx wi f(xi) = w f(x) a ≈ Z Xi=1 b b b nx ny nz 1 2 3 (x) (y) (z) f(x,y,z) dxdydz wi wj wk f(xi, yj,zk) a1 a2 a3 ≈ Z Z Z Xi=1 Xj=1 Xk=1 =(w(x) w(y) w(z))T f(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 F P = B (I B )(I B )(I B ) 16 16 16 2 ⊗ 8 4 ⊗ 4 8 ⊗ 2 10 1 0 0 1 0 ω4 B = ωn = exp( 2πi/n) 4 1 0 1 0 − − 0 1 0 ω − 4 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 1 1 W I if m> 1 m ⊗ 1 m ⊗ 1 W2m = − . [1] if m = 1 Fast Gauss Transform 2 g = exp( s t /δ) G = Gnear + G ij −k j − i k2 ⇒ far 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 Toeplitz blocks. 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).
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