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Chapter 5: Q50E (page 234)

a.Find all n脳nmatrices that are both orthogonal and upper triangular, with positive diagonal entries.

b.Show that the QRfactorization of an invertible n脳nmatrix is unique. Hint: If,A=Q1R1=Q2R2 thenthe matrixA=Q2-1Q1=Q2R1-1 is both orthogonal and upper triangular, with positive diagonal entries.

Short Answer

Expert verified
  1. n脳 nmatrices that are both orthogonal and upper triangular, with positive diagonal entries are

A =In

  1. Q1=Q2andlocalid="1659498166202" R1=R2

Step by step solution

01

Consider for part (a).

If A is an orthogonal and upper triangular matrix, thenA-1is lower and upper triangular becauseA-1=AT and it is an inverse of an upper triangular matrix. Thus,

AT=A-1

= A

Because A is orthogonal and diagonal with positive entries must be. So, A=In.

02

Consider for part (b).

Consider the theorem below.

Products and inverses of orthogonal matrices.

a.The product ABof two orthogonal n脳nmatrices Aand Bis orthogonal.

b.The inverse A-1 of an orthogonal n脳nmatrix Ais orthogonal.

Thus, according to the above theoremQ2-1Q1is orthogonal andR2R1-1is upper triangular with positive diagonal entries.

From the part (a) it has

Q2-1Q2=R1=R2

=Im

So that, Q1=Q2andR1=R2as claimed.

Hence,

  1. n脳 nmatrices that are both orthogonal and upper triangular, with positive diagonal entries will be

A=In

  1. Q1=Q2andR1=R2and both are orthogonal and upper triangular, with positive diagonal entries.

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Most popular questions from this chapter

Consider the vectors

ur=[11...1]andvr=[11...1]inRn

a.For n =2,3,4 , find the angle胃between role="math" localid="1659432008110" u鈯�and v鈯�. For n=2and 3, represent the vectors graphically.

b.Find the limit of 胃as napproaches infinity.

If A and B are arbitrary n脳nmatrices, which of the matrices in Exercise 21 through 26 must be symmetric?

BBT.

If then脳nmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?3A.

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

19.[212115]

Among all the vectors in Rnwhose components add up to 1, find the vector of minimal length. In the casen=2 , explain your solution geometrically.
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