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March 31

Source for calculation of singular vectors

I am looking for a step-by-step guide to calculating singular vectors. This is what I have so far: Given a matrix A, which is not square, calculate M as A matrix multiplied to A' (A transposed). It can be A'A or AA'. Does not matter for the final result. Calculate the eigenvectors of M as e1, e2, etc... ??? You have singular vectors. In attempting to fill in the ??? part, every web search shoves me to singular value decomposition. I am not attempting to calculate a singular value decomposition. I am trying to calculate the singular vectors of the original non-square matrix A.

For an example. I am attempting to calculate the singular vectors for the four data sets from Anscombe's quartet. For set 1, when I multiply A'A, I get the matrix [[1001, 797.6], [797.6, 660.17]]. I did the math and got the two eigenvalue,vector pairs: 1646.19 [1.24, 1] and 14.98 [-0.81,1]. I don't know what the singular vectors are. My understanding is the singular values are the square roots of the eigenvalues. So, the square root of 1646 is 40.57 and the square root of 14.98 is 3.87. But, I want the singular vectors, not the singular values. 68.187.174.155 (talk) 16:53, 31 March 2025 (UTC)[reply]