Transcription of 1 Singular values
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1 Singular values
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 . We compute that Ax 2= (Ax) (Ax) = (Ax)TAx=xTATAx=xT( x) = xTx= x Ax 2 0, it follows from the above equation that x 2 0.
Step 1. We rst need to nd the eigenvalues of ATA. We compute that ATA= 0 @ 80 100 40 100 170 140 40 140 200 1 A: We know that at least one of the eigenvalues is 0, because this matrix can have rank at most 2. In fact, we can compute that the eigenvalues are p 1 = 360, 2 = 90, and 3 = 0. Thus the singular values of Aare ˙ 1 = 360 = 6 p 10, ˙ 2 ...
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