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Chapter 5: Q49E (page 234)

Consider an invertible n脳nmatrix A. Can you write A=RQ, where Ris an upper triangular matrix and Q is orthogonal?

Short Answer

Expert verified

R=LT,Q=PTand A = RQ

Step by step solution

01

R is an upper triangular matrix.

It claimed that there exists upper triangular matrix R and an orthogonal matrix Q such that A= RQ.

Recall that, any n脳 nmatrix A with rank (A)=m, it is always possible to write A=QL where Q is an n脳 northogonal matrix and L is an n脳 nlower triangular matrix with positive diagonals.

Since, A is an invertible and so is ATand the rank of A is n.

It will find an orthogonal matrix P and a lower triangular matrix L,

AT=PL鈬�AT=PLT鈬�A=LTPTABT=BTAT

By observing in the above factorization. Since L is a lower triangular matrix therefore, LTwill be upper triangular matrix. Also, the matrix PT is orthogonal and hence we have a factorization for A.

Hence, R=LT,Q=PTand A = RQ is proved.

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