I need to find an invserse of the following matrix product:HT x W x H, whereH is an orthonormal matrix (the basis functions used in the columns of the matrix are orthonormal);W is a diagonal matrix with non-zero elements on the diagonal;and HT is the transpose of matrix H.I have a feeling that there has to be an easier way to find the inverse of this product then multipling these matrices first and then using LU Decomposition or Singular Value Decomposition.Any thoughts are appreciated.Thanks.

W x InvD x WT where InvD is the matrix of elements of the form 1/di

I'm mixing notation (in mine W is the orthogonal mat)...let me say it this way...generally, inverse of XY is Y inverse times X inverse....now Diag inverse is just the obvious, then use the fact orth have prop that W inv equals W transpose.

Thank you for your reply:The problem here is that H is not a square matrix (say it's n rows x k columns), and W is a square matrix (n x n).So the suggested H x InvW x HT is not a valid multiplication.Also, is the rule Inv(AB) = Inv(A) x Inv(B) general?I tried it on a simple example:A= [{1, 2}, {3, 4}] and B = [{5, 6}, {3, 5}]and Inv(A x B) <> Inv(A) x Inv(B).And finally, in my case the matrix is orthonormal, not orthogonal.Thanks

how can a matrix be diagonal if its not square?every orthonormal matrix is orthogonal.inv(AB) = inv(B)inv(A)MJ

Thanks for your comments.What I am trying to do is to solve the following system (determine vector z):(HT x W x H) x z = (HT x W x P), where P is a n x 1 vector, H is an n x k orthonormal matrix, and W is a n x n diagonal matrix with non-zero elements on the diagonal.the number of rows in H is about 100 and the number of columns is about 8.The product HT x W x H will result in a square matrix, without zero (or very close to zero) elements. Moreover, the matrix turns out to be symmetrical around the diagonal. Here is an example of what I mean:1 3 4 ..3 2 6 ..4 6 3 .... .. .. ..I am wandering if there is a computationally efficient way to solve this problem. Other than LU decomposition or SVD.Is there a better way to decompose a symmetrical matrix in a product of two triangular matrices?What is the best approach to take in this instance?Thanks