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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: Q 5.5-4E (page 260)

Question: In R^苍脳尘consider the inner product
$鈱� A, B 鈱� = t r a c e (A^{T} B)$
Defined in Example 3.
a.Find a formula for this inner product in R^苍脳1=Rⁿ
b.Find a formula for this inner product in R^1脳尘 (i.e., the space of row vectors with mcomponents).

Short Answer

Expert verified

Column vectors = $鈱�\overset{鈫�}{x}, \overset{鈫�}{y}鈱� = \overset{鈫�}{x} . \overset{鈫�}{y}$

Row vectors = $鈱�\overset{鈫�}{x}, \overset{鈫�}{y}鈱� = \overset{鈫�}{x} . \overset{鈫�}{y}$

Step by step solution

01

Consider for part (a).

An inner product on R^苍脳尘 , is $鈱�A, B鈱� = t r a c e (A^{T} B)$

Let $\overset{鈫�}{x} 鈭� R^{n 脳 1}$ be column vectors. Observe that

$\begin{array}{l} 鈱�\overset{鈫�}{x}, \overset{鈫�}{y}鈱� = t r a c e ({\overset{鈫�}{x}}^{T}, \overset{鈫�}{y}) \\ = t r a c e (\overset{鈫�}{x}, \overset{鈫�}{y}) \\ = \overset{鈫�}{x} . \overset{鈫�}{y} \end{array}$

02

Consider for part (b).

Let $\overset{鈫�}{x}, \overset{鈫�}{y}$ be two vectors. Write $\overset{鈫�}{x} = [x_{1}, x_{2}, ... x_{m}]$ and $\overset{鈫�}{y} = y_{1}, y_{2}, ... y_{m}]$ .

Observe that

${({\overset{鈫�}{x}}^{T}, \overset{鈫�}{y})}_{i j} x_{i} y_{j}$

So,

$\begin{array}{l} 鈱�\overset{鈫�}{x}, \overset{鈫�}{y}鈱� = t r a c e ({\overset{鈫�}{x}}^{T}, \overset{鈫�}{y}) \\ = {鈭�}_{i = 1}^{m} x_{i} y_{i} \\ = \overset{鈫�}{x} . \overset{鈫�}{y} \end{array}$

Hence,

For column vectors $鈱�\overset{鈫�}{x}, \overset{鈫�}{y}鈱� = \overset{鈫�}{x} . \overset{鈫�}{y}$
For row vectors $鈱�\overset{鈫�}{x}, \overset{鈫�}{y}鈱� = \overset{鈫�}{x} . \overset{鈫�}{y}$

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Most popular questions from this chapter

If Ais an $n 脳 m$ matrix, is the formula $im (A) = im ({AA}^{T})$ necessarily true? Explain.

TRUE OR FALSE? The equation $d e t (A^{T}) = d e t (A)$ holds for all $2 脳 2$ matrices A.

Consider a symmetric invertible n脳nmatrix Awhich admits an LDU-factorization A=LDU. See Exercises 90, 93, and 94 of Section 2.4. Recall that this factorization is unique. See Exercise 2.4.94. Show that

$U = L^{T}$ (This is sometimes called the $L D L^{T}$ - factorizationof a symmetric matrix A.)

a.Consider the matrix product $Q_{1} = Q_{2} S$ , where both $Q_{1}$ and $Q_{2}$ are n脳mmatrices with orthonormal columns. Show that Sis an orthogonal matrix. Hint: Computelocalid="1659499054761" ${Q_{1}}^{T} = Q_{1} = {(Q_{2} S)}^{T} Q_{2} S$ . Note that

${Q_{1}}^{T} = Q = {Q_{2}}^{T} = Q_{2} = I_{m}$

b.Show that the QRfactorization of an n脳mmatrix Mis unique. Hint: If $Q_{1} R_{1} = Q_{2} R_{2}$ , then $Q_{1} = Q_{2} R_{2} {R_{1}}^{- 1}$ . Now use part (a) and Exercise 50a.

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? $[\begin{matrix} 0.6 & 0.8 \\ 0.8 & 0.6 \end{matrix}]$ .

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