1 Singular values - University of California, Berkeley

Notes on Singular value decomposition for Math 54 Recall that ifAis a symmetricn nmatrix, thenAhas real eigenvalues 1,.., n(possibly repeated), andRnhas an orthonormal basisv1,..,vn,where each vectorviis an eigenvector ofAwith eigenvalue i. ThenA=PDP 1wherePis the matrix whose columns arev1,..,vn, andDis the diagonalmatrix whose diagonal entries are 1,.., n. Since the vectorsv1,..,vnareorthonormal, the matrixPis orthogonal, , so we can alternatelywrite the above equation asA=PDPT.(1)A Singular value decomposition (SVD) is a generalization of this whereAis anm nmatrix which does not have to be symmetric or even Singular valuesLetAbe anm nmatrix. Before explaining what a Singular value decom-position is, we first need to define the Singular values the matrixATA. This is a symmetricn nmatrix, so itseigenvalues are is an eigenvalue ofATA, then an eigenvector ofATAwith eigenvalue.

1 Singular values Let Abe an m nmatrix. Before explaining what a singular value decom-position is, we rst need to de ne the singular values of A. Consider the matrix ATA. This is a symmetric n nmatrix, so its eigenvalues are real. Lemma 1.1. If is an eigenvalue of ATA, then 0. Proof. Let xbe an eigenvector of ATAwith eigenvalue . We compute that

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