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Chapter 5: Q44E (page 225)

Consider an n脳mmatrix Awith rank(A) <m. Is it always possible to write A=QRWhere Qis an n脳mmatrix with orthonormal columns and Ris upper triangular? Explain.

Short Answer

Expert verified

No it is not possible.

Step by step solution

01

Definition of an Orthonormal matrix.

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

One way to express this is. Where QT is the transpose of Q and I is the identity matrix.

For example,

QTQ=q鈬赌1Tq鈬赌2Tq鈬赌3T..q鈬赌nTq鈬赌1,q鈬赌2,q鈬赌3......q鈬赌nIfm>n,then,

There is no n脳 mmatrix Q with orthonormal columns.

If the columns of a matrix are orthonormal, then they are linearly independent.

Hence, it is not possible to write A=QR.

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Most popular questions from this chapter

In Exercises 40 through 46, consider vectors in ; we are told that is the entry of matrix A.

A=[35115920112049]

46. Find , where V =span . Express your answer as a linear combination of and .

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

If then脳n matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?-B.

Consider the vector

v炉=[1234]in R4

Find a basis of the subspace of R4consisting of all vectors perpendicular to v鈫�.

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