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

symmetric

symmetric Ap.D.pt

man't AAT 7orthogonalpanddiagonalDst APDPgeneralizationAmin p.p.pt

A pDQTanymati diagonalmatrix aresquarerootsofthenonzeroeigenvaluesof Asymmetric

gsingularvalues

anevaluesarenonnegative

m.nlAInxnDr

fin Eyh hi ien

A Pmm DrmQtrmxn

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

evaluex datesawaexs.t.s xzisnot.nverte.ba

sa fstepFindeigenvalues Formatbasisof Nuts44 dme zxsm 111 s.ae YIIa I E 4 stepsAppiyastomatathe

stepandbasisforeach eigenvectorsorthonormal iiiupEffi 117eigenspan NoteItissutiastodoas 3Forart basisof NIGH withineaeneigenspan WIUJ f unit 117,433

111,1 wi ut III thisut ri

Ikea nT Gwi'ta.witg.wsConclusion S P Dp

t p.p.pt

SI's a.su ta sritcsswIwherePfIIEIEI.IE

s.ieis5IYanIEEaw andofiKiwi awi14GW's1c21514530treason All evaluesofSarepositive