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I'm working on this application where I get the spectral decomposition (O,D) of a matrix A for free (A is of full rank) but not A itself, and I need to only recover the last column of A. I was wondering if there was a more efficient way to do this than reconstructing the whole of A first?

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What is $D$? Diagonal matrix of eigenvalues? Do you have the eigenvector matrices for free too? – Jesse Chan Mar 13 '14 at 17:53
@JLC: I edited the question to avoid confusion! Thanks for the hint – user189035 Mar 13 '14 at 19:05
up vote 6 down vote accepted

Yes, for matrices A, B, the last column of matrix product A*B can be written as A*(the last column of B). You can use this fact to get the last column of the reconstruction using only O(N^2).

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