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Chapter 5: Q48E (page 234)

Consider an invertible n脳nmatrix A. Can you write Aas A=LQ, where Lis a lowertriangular matrix andQis orthogonal? Hint: Consider the QRfactorizationof AT.

Short Answer

Expert verified

L=RT,Q=PTandA=LQ

Step by step solution

01

L is a lower triangular matrix.

Given that A is an n脳n invertible matrix. We claimed that there exists lower triangular matrix L and an orthogonal matrix Q such that A=LQ.

SincedetA=detATand therefore,AT is also invertible. Now we have PR factorization for ,i.e.

There exists upper triangular matrix R and an orthogonal matrix P such that

AT=PR=(AT)T =(PR)T =A鈬�RTPT=(AB)T =BTAT

By observing in the above factorization. Since R is an upper triangular matrix therefore, will be lower triangular matrix. Also, the matrix PTis orthogonal and hence it has a factorization for A as lower triangular matrix and an orthonormal matrix.

Hence, L=RT,Q=PTandA=LQ.

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Most popular questions from this chapter

Consider a QRfactorization

M=QRShow that R=QTM.

Consider a symmetric n脳mmatrix A. What is the relationship between Im(A)and ker(A)?

a.Consider the matrix product Q1=Q2S, where both Q1and Q2 are n脳mmatrices with orthonormal columns. Show that Sis an orthogonal matrix. Hint: Computelocalid="1659499054761" Q1T=Q1=(Q2S)TQ2S. Note that

Q1T=Q=Q2T=Q2=Im

b.Show that the QRfactorization of an n脳mmatrix Mis unique. Hint: IfQ1R1=Q2R2, thenQ1=Q2R2R1-1 . Now use part (a) and Exercise 50a.

Find the least squarex鈬赌* of the systemAx鈬赌=b whereA=[123456789] andb鈬赌=[100] .

Find the length of each of the vectorsv鈬赌In exercises 1 through 3.

3.v鈬赌=[2345]

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