Applied regression analysis / Praktic / 1 / Literature / matrixid
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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) |
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(A + B)T = AT + BT |
(1b) |
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(AB)T = BT AT |
(1c) |
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if individual inverses exist (AB) 1 |
= B 1A 1 |
(1d) |
(A 1)T |
= (AT ) 1 |
(1e) |
0.2 trace, determinant and rank
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jABj = jAjjBj |
(2a) |
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jA 1j |
1 |
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= |
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(2b) |
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jAj |
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jAj |
= Yevals |
(2c) |
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Tr [A] |
= evals |
(2d) |
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if the cyclic products are well de ned; |
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Tr [ABC : : :] = Tr [BC : : : A] = Tr [C : : : AB] = : : : |
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(2e) |
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rank [A] = rank |
AT A = rank |
AAT |
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(2f) |
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biggest eval |
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condition number = = r |
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(2g) |
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smallest eval |
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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.
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0.3derivatives of traces
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@Tr [X] |
= I |
(3a) |
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@X |
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@Tr [XA] |
= |
@Tr [AX] |
= AT |
(3b) |
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@X |
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@X |
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@Tr XT A |
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@Tr AXT |
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= A |
(3c) |
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@X |
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@X |
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XT AX |
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@Tr @X |
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= (A + AT )X |
(3d) |
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@Tr X 1A |
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@X |
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= X 1AT X 1 |
(3e) |
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0.4derivatives of determinants
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@jAXBj |
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AXB |
(X 1)T = |
AXB |
(XT ) 1 |
(4a) |
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@X |
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@ ln jXj |
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= (X 1)T |
= (XT ) 1 |
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(4b) |
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@X |
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@ ln jX(z)j |
= Tr |
X 1 |
@X |
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@z |
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@z |
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@jXT AXj |
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XT AX |
(AX(XT AX) 1 |
+ AT X(XT AT X) 1) (4d) |
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0.5derivatives of scalar forms
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@(aT x) |
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@(xT a) |
= a |
(5a) |
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@x |
@x |
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@(xT Ax) |
= (A + AT )x |
(5b) |
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@x |
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@(aT Xb) |
= abT |
(5c) |
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@X |
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@(aT XT b) |
= baT |
(5d) |
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@X |
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@(aT Xa) |
@(aT XT a) |
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= |
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= aaT |
(5e) |
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@X |
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@X |
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@(aT XT CXb) |
= CT XabT + CXbaT |
(5f) |
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@X |
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@ (Xa + b)T C(Xa + b) |
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@X |
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= (C + CT )(Xa + b)aT |
(5g) |
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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
R
represents the ratio of the hypervolume dy to that of dx so that |
f(y)dy = |
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f(y(x))j@y |
T |
=@xjdx. |
T |
=@x |
T |
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however, the sloppy forms @y=@x, @yR |
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@y=@xT are often used for this Jacobain matrix.
0.6derivatives of vector/matrix forms
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@(X 1) |
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@X |
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= X 1 |
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X 1 |
(6a) |
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@z |
@z |
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@(Ax) |
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= A |
@x |
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(6b) |
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@z |
@z |
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@(XY) |
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= X |
@Y |
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@X |
Y |
(6c) |
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@z |
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@z |
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@(AXB) |
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@X |
B |
(6d) |
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@z |
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@z |
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@(xT A) |
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(6e) |
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@x |
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@(xT ) |
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(6f) |
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@x |
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@(xT AxxT ) |
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= (A + AT )xxT + xT AxI |
(6g) |
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0.7constrained maximization
the maximum over x of the quadratic form:
T x |
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2xT A 1x |
(7a) |
subject to the J conditions cj(x) = 0 is given by:
A + AC ; = 4(CT AC)CT A |
(7b) |
where the jth column of C is @cj(x)=@x
0.8symmetric matrices
have real eigenvalues, though perhaps not distinct and can always be diagonalized to the form:
A = C CT |
(8) |
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where the columns of C are (orthonormal) eigenvectors (i.e. CCT = I) and the diagonal of has the eigenvalues
0.9block 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 |
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A21 |
A221 |
= jA22j jF11j = jA11j jF22j |
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A |
A |
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F 1 |
A 1A F 1 |
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A21 |
A22 |
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F221A21A111 |
F221 |
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12 22 |
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A111 + A111A12F221A21A111 |
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F111A12A221 |
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where |
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A221A21F111 |
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A221 + A221A21F111A12A221 |
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F11 = A11 A12A221A21 |
F22 = A22 A21A111A12 |
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for block diagonal matrices things are much easier:
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A011 |
A0221 |
= jA11jjA22j |
(9d) |
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A 1 0 |
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0 A221 |
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0.10matrix inversion lemma (sherman-morrison-woodbury)
using the above results for block matrices we can make some substitutions and get the following important results:
(A + XBXT ) 1 = A 1 A 1X(B 1 + XT A 1X) 1XT A 1 |
(10) |
jA + XBXT j = jBjjAjjB 1 + XT A 1Xj |
(11) |
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 converted 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
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