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Sam Roweis' Notes on Matrix Identities

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Sam Roweis' Notes on Matrix Identities matrix identities sam roweis (revised June 1999) note that a,b,c and A,B,C do not depend on X,Y,x,y or z 0.1 basic formulae A(B + C) = AB + AC (1a) (A + B)T = AT + BT (1b) (AB)T = BT AT (1c) 1 1 1 if individual inverses exist (AB)− = B− A− (1d) 1 T T 1 (A− ) = (A )− (1e) 0.2 trace, determinant and rank AB = A B (2a) j j j jj j 1 1 A− = (2b) j j A j j A = evals (2c) j j Y Tr [A] = evals (2d) X if the cyclic products are well defined; Tr [ABC :::] = Tr [BC ::: A] = Tr [C ::: AB] = ::: (2e) rank [A] = rank AT A = rank AAT (2f) biggest eval condition number = γ = r (2g) smallest eval derivatives of scalar forms with respect to scalars, vectors, or matricies are indexed in the obvious way. similarly, the indexing for derivatives of vectors and matrices with respect to scalars is straightforward. 1 0.3 derivatives of traces @Tr [X] = I (3a) @X @Tr [XA] @Tr [AX] = = AT (3b) @X @X @Tr XT A @Tr AXT = = A (3c) @X @X @Tr XT AX = (A + AT )X (3d) @X 1 @Tr X− A 1 T 1 = X− A X− (3e) @X − 0.4 derivatives of determinants @ AXB 1 T T 1 j j = AXB (X− ) = AXB (X )− (4a) @X j j j j @ ln X j j = (X 1)T = (XT ) 1 (4b) @X − − @ ln X(z) 1 @X j j = Tr X− (4c) @z @z T @ X AX T T 1 T T T 1 j j = X AX (AX(X AX)− + A X(X A X)− ) (4d) @X j j 0.5 derivatives of scalar forms @(aT x) @(xT a) = = a (5a) @x @x @(xT Ax) = (A + AT )x (5b) @x @(aT Xb) = abT (5c) @X @(aT XT b) = baT (5d) @X @(aT Xa) @(aT XT a) = = aaT (5e) @X @X @(aT XT CXb) = CT XabT + CXbaT (5f) @X @ (Xa + b)T C(Xa + b) = (C + CT )(Xa + b)aT (5g) @X 2 the derivative of one vector y with respect to another vector x is a matrix whose (i; j)th element is @y(j)=@x(i). such a derivative should be written as @yT =@x in which case it is the Jacobian matrix of y wrt x. its determinant represents the ratio of the hypervolume dy to that of dx so that f(y)dy = f(y(x)) @yT =@x dx. however, the sloppy forms @y=@x, @yRT =@xT and @Ry=@xT arej oftenj used for this Jacobain matrix. 0.6 derivatives of vector/matrix forms 1 @(X− ) 1 @X 1 = X− X− (6a) @z − @z @(Ax) @x = A (6b) @z @z @(XY) @Y @X = X + Y (6c) @z @z @z @(AXB) @X = A B (6d) @z @z @(xT A) = A (6e) @x @(xT ) = I (6f) @x @(xT AxxT ) = (A + AT )xxT + xT AxI (6g) @x 0.7 constrained maximization the maximum over x of the quadratic form: T 1 T 1 µ x x A− x (7a) − 2 subject to the J conditions cj(x) = 0 is given by: Aµ + ACΛ; Λ = 4(CT AC)CT Aµ (7b) − where the jth column of C is @cj(x)=@x 0.8 symmetric matrices have real eigenvalues, though perhaps not distinct and can always be diag- onalized to the form: A = CΛCT (8) 3 where the columns of C are (orthonormal) eigenvectors (i.e. CCT = I) and the diagonal of Λ has the eigenvalues 0.9 block matrices for conformably partitioned block matrices, addition and multiplication is performed by adding and multiplying blocks in exactly the same way as scalar elements of regular matrices however, determinants and inverses of block matrices are very tricky; for 2 blocks by 2 blocks the results are: A11 A12 = A22 F11 = A11 F22 (9a) A21 A22 j j · j j j j · j j 1 1 1 1 A11 A12 − F11− A11− A12F22− = 1 1 − 1 (9b) A21 A22 F− A21A− F− − 22 11 22 1 1 1 1 1 1 A11− + A11− A12F22− A21A11− F11− A12A22− = 1 1 1 − 1 1 1 A− A21F− A− + A− A21F− A12A− − 22 11 22 22 11 22 where 1 1 F11 = A11 A12A− A21 F22 = A22 A21A− A12 − 22 − 11 for block diagonal matrices things are much easier: A11 0 = A11 A22 (9d) 0 A22 j jj j 1 1 A11 0 − A11− 0 = 1 (9e) 0 A22 0 A22− 0.10 matrix inversion lemma (sherman-morrison-woodbury) using the above results for block matrices we can make some substitutions and get the following important results: T 1 1 1 1 T 1 1 T 1 (A + XBX )− = A− A− X(B− + X A− X)− X A− (10) − T 1 T 1 A + XBX = B A B− + X A− X (11) j j j jj jj j where A and B are square and invertible matrices but need not be of the same dimension. this lemma often allows a really hard inverse to be con- verted into an easy inverse. the most typical example of this is when A is large but diagonal, and X has many rows but few columns 4.
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