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Chapter 5: Q68E (page 235)

The formulaA(ATA)-1 for the matrix of an orthogonalprojection is derived in Exercise 67. Now considerthe QRfactorization of A, and express the matrixA(ATA)-1ATin terms of Q.

Short Answer

Expert verified

The equationQTQ=Im holds.

Step by step solution

01

Consider the theorem below.

The matrix of an orthogonal projection:Consider a subspace VofRnwith orthonormal basis role="math" localid="1659500909794" u鈫�1,u鈫�2,.......u鈫�m. The matrix Pof the orthogonal projection onto Vis

P=QQTWhere Q=[|||u鈫�1u鈫�2...u鈫�m|||].

Pay attention to the order of the factors ( QQTas opposed to QTQ). Note that matrix Pis symmetric, since

PT=(QQT)T=(QT)TQT=QQT=P

So, if A = QR , then,

AATA-1A=QRRTQTQR-1RTQT=QRRTR-1RTQT=QRR-1RT-1RTQT=QQT

As mentioned in the above theorem.

Hence, the equation QTQ=Imholds since the columns of Q are orthonormal.

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Most popular questions from this chapter

Consider a consistent system Ax鈫�=b鈫�.

(a) Show that this system has a solution x鈫�0 in (kerA)鈯� .

(b) Show that the systemAx鈫�=b鈫� has only one solution in (kerA)鈯� .

(c) Ifx鈫�0 is the solution in (kerA)鈯� androle="math" localid="1660124695419" x鈫�1is another solution of the system Ax鈫�=b鈫� , show that||x鈫�0||<||x鈫�1|| . The vectorx鈫�0 is called the minimal solution of the linear system Ax鈫�=b鈫� .

Consider an n x m matrix A with rank (A) = m. Is it always possible to write A as A = QL where Q is an n x m matrix with orthonormal columns and L is a lower triangular m x m matrix with positive diagonal entries? Explain.

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

(a) Consider an matrix A such that A螕A=Im. It is necessarily true that? Explain.

(b) Consider an n脳nmatrix A such that ATA=In. Is it necessarily true that AAT=In? Explain.

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