MATLAB: Efficiently calculating the trace of a matrix product

MATLABmatrix manipulationmatrix multiplicationtrace

I have two NxN square matrices, A and B, and I would like to calculate the trace of AB. Since the trace of AB only depends on its diagonal elements, it should hypothetically not be necessary to compute all of AB, thereby reducing the amount of operations from N^3 to N^2. My question is twofold:

Does calling tr(AB) in MATLAB automatically exploit this fact?
If not, is there an efficient way of doing this that doesn't involve calling for loops?

Thanks!

Best Answer

Bt=B.';
traceProduct = A(:).'*Bt(:);

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MATLAB: Multiply then sum elements of two matrices

It is not clear from your question whether corresponding elements are to be multiplied or if you are wanting to do a matrix multiplication.

If corresponding elements are to be multiplied, then you could calculate the sum-of-products using

dot(A(:),B(:))

MATLAB: Symmetric Matrix-Vector Multiplication with only lower triangular stored

Plese check if this is efficient in your case:

A = rand(5, 5);
A = A + A';      % Symmetric example matrix
B = tril(A);     % Left triangular part
x = rand(5, 1);
y1 = A * x;
y2 = B * x + B.' * x - diag(B) .* x;
y1 - y2
ans = 5×1
	1.0e+-15 *

         0
         0
   -0.4441
         0
         0

I assume that Matlab's JIT can handle B.' * x without computing B.' explicitly. Alternative:

y2 = B * x + B.' * x - diag(diag(B)) * x;

Best Answer

Related Solutions

MATLAB: Multiply then sum elements of two matrices

MATLAB: Symmetric Matrix-Vector Multiplication with only lower triangular stored

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