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Worksheet 24 (April 9)
DIS 119/120 GSI Xiaohan Yan
1 Review
DEFINITIONS
• symmetric matrices, orthogonally diagonalizable;
• singular value (of non-square matrices).
METHODS AND IDEAS
Theorem 1. (Properties of Symmetric Matrices)Let A be a symmetric n ˆ n matrix, then:(a) all eigenvalues of A are real;(b) eigenspaces of di↵erent eigenvalues of A are orthogonal;(c) there is a basis of Rn consisting of orthonormal eigenvectors of A (whichmeans A is orthogonally diagonalizable).
In fact the converse of (c) is also true: if a square matrix is orthogonallydiagonalizable, it must be symmetric, too.
Method 1. (Algorithm of Singular Value Decomposition)Assume that we are given an m ˆ n matrix A with m ° n.
1 Find eigenvalues of the symmetric matrix ATA, which are all real andnon-negative. Denote them by �1, ¨ ¨ ¨ ,�n. Assume that the first r of themare strictly positive and the rest are zero.
2 Find orthonormal eigenvectors of ATA corresponding to the strictly posi-tive eigenvalues �1, ¨ ¨ ¨ ,�r. Denote them by v1, ¨ ¨ ¨ ,vr.
3 Let �i “ ?�i for i “ 1, ¨ ¨ ¨ , r. These are the singular values of A.
4 Let ui “ 1�iAvi for i “ 1, ¨ ¨ ¨ , r. Then,
A “ �1u1vT1 ` ¨ ¨ ¨ ` �rurv
Tr .
This is the singular value decomposition(SVD) of A.
Note that in this case rankA “ r. Moreover, tv1, ¨ ¨ ¨ ,vru is an orthonormalbasis of RowpAq, while tu1, ¨ ¨ ¨ ,uru is an orthonormal basis of ColpAq.
1
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2 Problems
Example 1. Diagonalize the matrix below with orthogonal matrices.
S Ҭ
˝2 1 11 2 11 1 2
˛
‚.
Example 2. Let us still consider the matrix S from the previous example. Isthere a vector x P R3 such that xTSx † 0?
Example 3. Find SVD of the matrix
A Ҭ
˝1 32 23 1
˛
‚.
2
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