LA-I's SVD note stated and sketched its connection to the eigenvalues of , but explicitly deferred a full existence proof, citing that LA-I's symmetric-matrix eigenstructure only covered the distinct-eigenvalue case. That deferral is resolved: Topic 4 proved the spectral theorem in complete generality, and this topic uses it to construct the SVD for every matrix, with no restriction on shape, rank, or repeated singular values whatsoever.
By the end of this topic you should be able to:
- construct the SVD of any matrix from the spectral decomposition of ;
- explain precisely how is built once is known;
- state the SVD's geometric picture in terms of the four fundamental subspaces made orthonormal;
- construct and interpret the polar decomposition.
In this lesson
- Contents
- The big picture
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