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Chapter 5: Q47E (page 234)

If A=QRis a QRfactorization, what is the relationship between ATA and RTR?

Short Answer

Expert verified

ATA=RTR

Step by step solution

01

QR factorization

Consider an n脳mmatrix Mwith linearly independent column v鈯�1,....,v鈯�m.Then there exists an n脳mmatrix Qwhose columns u11,....,u1mare orthonormal and an upper triangular matrix Rwith positive diagonal entries such that.

M=QR

To find the relation between ATAandRTRconsider.

Since, the column of the matrix Q is orthonormal and therefore it has QQT=l=QTQ. Thus, it is find as

AAA=QRTQR=RTQTQR=RTQTQR=RTIR=RTR

Hence, the relation between ATAand RTRis ATA=RTR.

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Most popular questions from this chapter

TRUE OR FALSE? The equation det(AT)=det(A)holds for all 2脳2matrices A.

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -B.

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u鈫�=[234]andv鈫�[1k1]and perpendicular?

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.

TRUE OR FALSE?If matrices A and Sare orthogonal, then S-1ASis orthogonal as well.
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