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Chapter 5: Q46E (page 234)

Consider a QRfactorization

M=QRShow that R=QTM.

Short Answer

Expert verified

R=QTM

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 columnsu11,....,u1mare orthonormal and an upper triangular matrix Rwith positive diagonal entries such that.

M=QR

To prove thatR=QTM

Since, the column of the matrix Q is orthonormal and therefore it gives,

QQT=l=QTQ

Thus, it is written as,

M=QR鈬�QTM=QQR鈬�QTM=IR鈬�R=QTM

Hence, R=QTMproved.

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Most popular questions from this chapter

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

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