# Matrix Identities

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### Cauchy-Schwarz Inequality

• |xHy|2 = xHyyHx <= xHxyHy for any complex vectors x and y[4.16]
• XHY(YHY)-1YHX <= XHX where <= represents the Loewner partial order[4.17]

### Convolution Identities

conv(a[m],b[n])[m+n-1] is the convolution of a and b,. The r'th element is conv(a[m],b[n])r = ap:qT(Jb)r-q+1:r-p+1where p=max(1,r-n+1) and q=min(r,m).

• conv(a,b) = conv(b,a)
• conv(a+b,c)=conv(a,c)+conv(b,c)
• conv([a[p] ; b[q]],[c[r] ; d[s]]) = [conv(a,c); 0[q+s]] + [0[p]; conv(b,c); 0[s]] + [0[r]; conv(a,d); 0[q]] + [0[p+r]; conv(b,d)]
• conv([a[p] ; b],[c[r] ; d]) = [conv(a,c); 0; 0] + [0[p]; bc; 0] + [0[r]; da; 0] + [0[p+r]; bd]
• conv([a[m] ; b[n]],c) = [conv(a,c); 0[n]] + [0[m]; conv(b,c)]
• conv([a[m] ; b],c) = [conv(a,c); 0] + [0[m]; bc]
• conv([a ; 0[n]],c) = [conv(a,c); 0[n]]
• conv([0[m] ; b],c) = [0[m]; conv(b,c)]
• conv(Ja,Jb)=J conv(a,b)
• TOE(a[m]) b[n] = conv(a,b)n:m

### Durbin Recursion

If x satisfies R[n#n] x = [1; 0] and R is a symmetric toeplitz matrix and hence of the form TOE([Jr; 0]+[0; r])[n#n] for some r[n], where J is the exchange matrix. Then

• We can find y to solve Sy = [1; 0] where
• S is the augmented symetric toeplitz matrix given by TOE([s; Jr; 0]+[0; r; s])[n+1#n+1]
• k = [r; s]T J [x; 0]
• y[n+1] = ([x; 0] - k J [x; 0])/(1-k^2)

This recursion gives an efficient way of solving equations of the form R x = [1; 0].

### Inversion Sums

• A + B = A (A-1 + B-1) B = B (A-1 + B-1) A = A (A-1 + A-1BA-1) A = B (B-1 + B-1AB-1) B
• A + I = A (A-1 + I) = (A-1 + I) A
• A-1 + B-1 = A-1 (A + B) B-1 = B-1 (A+ B) A-1 = A-1 (A + AB-1A) A-1 = B-1 (B+ BA-1B) B-1
• A-1 + I = A-1 (A + I) = (A + I) A-1
• AB + BC = A (A-1B + BC-1) C
• (A-1 + B-1)-1 = A (A + B)-1B = B (A + B)-1A = A - A (A + B)-1A = B - B (A + B)-1B
• (A-1 + I)-1 = A (A + I)-1 = (A + I)-1A = A - A (A + I)-1A = I - (A + I)-1

### Inversion Identities

These identities are useful because it says how a matrix changes if you add a bit onto its inverse. They are variously called the Matrix Inversion Lemma, Sherman-Morrison formula and Sherman-Morrison-Woodbury formula.

• [ (I+VHAU) non-singular]: (A-1 + UVH)-1 = A - AU(I + VHAU)-1VHA     [4.2]
• [ (I+VHA-1U) non-singular]: (A + UVH)-1 = A-1 - A-1U(I + VHA-1U)-1VHA-1
• [vHAu != -1]: (A-1 + uvH)-1 = A - AuvHA/(1 + vHAu)
• [ (C+VHAU) non-singular]: (A-1 + UC-1VH)-1 = A - AU(C + VHAU)-1VHA
• [ (C+VHAV) non-singular]: (A-1 + VC-1VH)-1 = A - AV(C + VHAV)-1VHA
• [ (I+VHAV) non-singular]: (A-1 + VVH)-1 = A - AV(I + VHAV)-1VHA
• [ (I+VHAU) non-singular]: (A-1 + UVH)-1U = AU(I + VHAU)-1
• [ (I+VHAU) non-singular]: VH(A-1 + UVH)-1 = (I + VHAU)-1VHA
• [ (I+VHAU) non-singular]: VH(A-1 + UVH)-1U = I-(I + VHAU)-1
• det(A-1 + UVH) = det(I + VHAU) × det(A-1) [4.3] sometimes called the Matrix Determinant Lemma
• det(A-1 + uvH) = (1 + vHAu) × det(A-1)

### Projection Matrix Identities

These identities show how the matrix that projects onto the column-space of X changes if extra columns are added to X. We define the projection matrices P(X)=X(XHX)#XH and Q(X)=I-P(X), these respectively project onto the column space and null space of X.

• P([X Y]) = P(X) + P(Q(X)Y) = P(X) + Q(X) Y (YH Q(X) Y)# YH Q(X)
• [yHQ(X)y = 0] P([X y]) = P(X)
• [yHQ(X)y != 0] P([X y]) = P(X) + P(Q(X)y) = P(X) + Q(X) y yH Q(X) / (yH Q(X) y)
•
• [yHQ(X)y != 0] zHP([X y])z = zHP(X)z + |zHQ(X) y|2 / (yH Q(X) y)
• Q([X Y]) = Q(X) - Q(X) Y (YH Q(X) Y)# YH Q(X)

### Toeplitz and Hankel Identities

If a vector is multiplied by a triangular toeplitz  matrix, it is possible to exchange the components of the vector and matrix. All matrices below are n#n.

• TOE([0[n-1]; p[n]])q = TOE([0[n-1]; q[n]])p = conv(p,q)1:n
• TOE([ p[n]; 0[n-1]])q = TOE([ q[n]; 0[n-1]])p = conv(p,q)n:2n-1

Similar expressions may be obtained for hankel matrices by observing that if T is toeplitz then TJ is hankel and Tp=(TJ)(Jp)

In the identity below, the matrix on the left of the = is a symmetric toeplitz matrix with 2a1 on the main diagonal. The matrix on the right is the sum of a lower triangular toeplitz and an upper triangular hankel matrix and has 2b as its first column.

• TOE([Ja[n]; 0[n-1]]+[0[n-1]; a[n]])[n#n]b = (TOE([b[n]; 0[n-1]])[n#n]J+TOE([0[n-1]; b[n]])[n#n])a .

This page is part of The Matrix Reference Manual. Copyright © 1998-2021 Mike Brookes, Imperial College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "imperial.ac.uk".
Updated: \$Id: identity.html 11291 2021-01-05 18:26:10Z dmb \$