Q68E The formula聽A(ATA)-1 for the ma... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q68E (page 235)

The formula $A {(A^{T} A)}^{- 1}$ for the matrix of an orthogonalprojection is derived in Exercise 67. Now considerthe QRfactorization of A, and express the matrix $A {(A^{T} A)}^{- 1} A^{T}$ in terms of Q.

Short Answer

Expert verified

The equation $Q^{T} Q = I_{m}$ holds.

Step by step solution

Consider the theorem below.

The matrix of an orthogonal projection:Consider a subspace Vof $R^{n}$ with orthonormal basis role="math" localid="1659500909794" ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}, . . . . . . . {\overset{鈫�}{u}}_{m}$ . The matrix Pof the orthogonal projection onto Vis

$P = Q Q^{T}$ Where $Q = [\begin{matrix} | & | & | \\ {\overset{鈫�}{u}}_{1} & {\overset{鈫�}{u}}_{2} . . . & {\overset{鈫�}{u}}_{m} \\ | & | & | \end{matrix}]$ .

Pay attention to the order of the factors ( $Q Q^{T}$ as opposed to $Q^{T} Q$ ). Note that matrix Pis symmetric, since

$P^{T} = {(Q Q^{T})}^{T} = {(Q^{T})}^{T} Q^{T} = Q Q^{T} = P$

So, if A = QR , then,

$A {(A^{T} A)}^{- 1} A = Q R (R^{T} Q^{T} Q R^{- 1}) R^{T} Q^{T} = Q R (R^{T} R^{- 1}) R^{T} Q^{T} = Q R R^{- 1} {(R^{T})}^{- 1} R^{T} Q^{T} = Q Q^{T}$

As mentioned in the above theorem.

Hence, the equation $Q^{T} Q = I_{m}$ holds since the columns of Q are orthonormal.

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91影视

The formula $A {(A^{T} A)}^{- 1}$ for the matrix of an orthogonalprojection is derived in Exercise 67. Now considerthe QRfactorization of A, and express the matrix $A {(A^{T} A)}^{- 1} A^{T}$ in terms of Q.

Short Answer

Step by step solution

Consider the theorem below.

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91影视

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

Step by step solution

Consider the theorem below.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Geometry

Calculus

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

The formula $A {(A^{T} A)}^{- 1}$ for the matrix of an orthogonalprojection is derived in Exercise 67. Now considerthe QRfactorization of A, and express the matrix $A {(A^{T} A)}^{- 1} A^{T}$ in terms of Q.