More specifically, A = USV^{t} where A is a square positive definite matrix with all non-diagonal entries equal to 0. In this case, S (matrix of singular values) is just the sorted diagonal values of A; U and V are made up of 1 and -1, but I can't seem to figure out if there's a quick way to figure out which entries of U and V are 1 vs 0.
Is there a shortcut to deriving U and V when doing SVD on a diagonal matrix?
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linear-algebra
matrix-decomposition
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1 Answers
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Note that in general, SVD is not unique. However, a way to find one possible $U,V$ for your matrix is as follows:
Let $d_1,d_2,\dots,d_n$ denote the diagonal entries of $A$. Suppose that the singular values are $d_{i_1},d_{i_2},\dots,d_{i_n}$ where $\{i_1,\dots,i_n\}$ is a rearrangement of $1, \dots, n$. Let $e_1,\dots,e_n$ denote the columns of the identity matrix (the standard basis). It suffices to take $U = V$ where the $k$ column of $U$ is $e_{i_k}$.
For example, with $$ A = \pmatrix{1\\&4\\&&2} $$ take $$ U = V = \pmatrix{0&0&1\\1&0&0\\0&1&0} $$