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Chapter 5: Q20E (page 263)

Every invertible matrix Acan be expressed as the product of an orthogonal matrix and an upper triangular matrix.

Short Answer

Expert verified

True

Step by step solution

01

Step by step solution Step 1: Consider the theorem below.

QR factorization: Consider an n脳mmatrix Mwith linearly independent columns v鈫�1,....,v鈫�m. Then there exists an n脳mmatrix Qwhose columnsu鈫�1,....,u鈫�m are orthonormal and an upper triangular matrix Rwith positive diagonal entries such thatM=QR.

02

Determine whether the statement is true or false

Furthermore r11=||v鈫�1||,rjj=||v鈫�j鈯�||

(j=2,....,m) and rij=u鈫�i.v鈫�j

Since, the matrix is invertible hence its columns will be linearly independent and therefore, using QRdecomposition,there exists orthogonal matrix Q and an upper triangular matrix R such thatA=QR.

Hence, the statement is true.

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Most popular questions from this chapter

Consider the orthonormal vectors u鈫�1,u鈫�2,u鈫�3,u鈫�4,u鈫�5 in鈩�10. Find the length of the vectorx鈫�=7u鈫�1-3u鈫�2+2u鈫�3+u鈫�4-u鈫�5.

Question: If the n脳nmatrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A+B.

If then脳n matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?A+B.

If the nxnmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well? AT.

Prove Theorem 5.1.8d.(V鈯�)鈯�=Vfor any subspaceV of鈩�n.
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