eig versus svd functions?
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Hi,
I would like to ask what is the difference between the function eig and svd and what is the difference between the right eigenvectors and the right singular eigenvectors of these functions?
Thank you
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Alfonso Nieto-Castanon
2014-7-18
编辑:Alfonso Nieto-Castanon
2014-7-18
SVD is a decomposition for arbitrary-size matrices, while EIG applies only to square matrices. They are very much related:
The right singular vectors of A are the eigenvectors of A'*A, and the left singular vectors of A are the eigenvectors of A*A'.
Similarly the singular values of A are the square root of the eigenvalues of A*A' (or A'*A, the eigenvalues of those are just the same)
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Alfonso Nieto-Castanon
2014-7-18
not exactly, there are simply no "eigenvectors" of a non-square matrix (eigenvalues/eigenvectors are only defined for square matrices)
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Traian Preda
2014-7-18
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Alfonso Nieto-Castanon
2014-7-18
the eigenvectors of a square matrix are not generally the same as any of the singular vectors of that same matrix (they are equal/equivalent only when the matrix is symmetric)
Traian Preda
2014-7-18
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Alfonso Nieto-Castanon
2014-7-18
编辑:Alfonso Nieto-Castanon
2014-7-18
You can reconstruct A from its eigenvectors only if A is normal (A'*A==A*A'). You can reconstruct A from its singular vectors for any matrix A.
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