matlab,matrix,least-squares , How to solve for matrix in Matlab?


How to solve for matrix in Matlab?

Question:

Tag: matlab,matrix,least-squares

How can I solve eq1, where eq2 and eq3 and eq4 eq5 in the least squares sense in matlab?

So I'd like to have the minimizing eq6 as output.


Answer:

Rewrite the quantity to minimise as

||Xa - b||^2

= (definition of the Frobenius norm)

Tr{(Xa - b) (Xa - b)'}

= (expand matrix-product expression)

Tr{Xaa'X' - ba'X' - Xab' + bb'}

= (linearity of the trace operator)

Tr{Xaa'X'} - Tr{ba'X'} - Tr{Xab'} + Tr{bb'}

= (trace of transpose of a matrix = trace of the matrix)

Tr{Xaa'X'} - 2 Tr{ba'X'} + Tr{bb'}

where ' denotes the transpose operator (because all matrices involved are real, transpose and conjugate transpose are the same). Now, if you refer to section 2.5 of the Matrix Cookbook, you'll find that

(All derivatives are with respect to matrix X).

Therefore, the matrix that minimises the quantity of interest satisfies

2Xaa' = 2ba'
Xaa' = ba'

Therefore, you can use MATLAB's matrix right-division operator, /, to compute X:

X = b * a' / (a * a');

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