Matrix Identities

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Cauchy-Schwarz Inequality


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).

See also: Toeplitz for more identities relating conv() and TOE()


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

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


Inversion Sums


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.


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.


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.

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.


This page is part of The Matrix Reference Manual. Copyright © 1998-2005 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".
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