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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 8: Q29E (page 393)

Consider a symmetric matrixA. If the vector $\overset{鈬赌}{v}$ is in the image of Aand $\overset{鈬赌}{w}$ is in the kernel of A, is $\overset{鈬赌}{v}$ necessarily orthogonal to $\overset{鈬赌}{w}$ ? Justify your answer.

Short Answer

Expert verified

The $\overset{鈬赌}{v}$ is orthogonal to $\overset{鈬赌}{w}$ .

Step by step solution

01

The matrix

A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an item in mathematics.
For example, is a two-row, three-column matrix

02

Determine the orthogonal matrix

A theorem we studied in this book states that

${(im A)}^{-} = \ker (A^{T})$

And we know that A is symmetric $鈬� A = A^{T}$ , which implies

${(im A)}^{-} = \ker (A^{T}) 鈫� {(im A)}^{-} = \ker (A)$

Hence $\overset{鈬赌}{v}$ is orthogonal to $\overset{鈬赌}{w}$ .

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Most popular questions from this chapter

The determinant of a negative definite $4 脳 4$ matrix must be positive.

For the matrix $A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}],$ write $A = B^{2}$ as discussed in Exercise 30. See Example 1.

Let be a complex $n 脳 n$ matrix such that $| 位 | < 1$ for all eigenvalues $位$ of . Show that role="math" localid="1659610526426" $\underset{t 鈫� 鈭�}{l i m} A^{t} = 0$ , meaning that the modulus of all entries of $A^{t}$ approaches zero.

b. Prove Theorem 7.6.2.

A Cholesky factorization of a symmetric matrix A is a factorization of the form $A = L L^{T}$ where L is lower triangular with positive diagonal entries.

Show that for a symmetric $n 脳 n$ matrix A, the following are equivalent:

(i) A is positive definite.

(ii) All principal submatricesrole="math" localid="1659673584599" $A^{(m)}$ of A are positive definite. See

Theorem 8.2.5.

(iii) $\det (A^{(m)}) > 0 for m = 1, . . . ., n$

(iv) A has a Cholesky factorization $A = L L^{T}$

Hint: Show that (i) implies (ii), (ii) implies (iii), (iii) implies (iv), and (iv) implies (i). The hardest step is the implication from (iii) to (iv): Arguing by induction on n, you may assume that $A^{(n - 1)}$ ) has a Cholesky factorization $A^{(n - 1)} = {BB}^{T}$ . Now show that there exist a vector $\overset{鈫�}{x} i n R^{n - 1}$ and a scalar t such that

$A = [\begin{matrix} A^{(n - 1)} & \overset{鈫�}{v} \\ {\overset{鈫�}{v}}^{T} & k \end{matrix}] = [\begin{matrix} B & 0 \\ {\overset{鈫�}{x}}^{T} & 1 \end{matrix}] [\begin{matrix} B^{T} & \overset{鈫�}{x} \\ 0 & t \end{matrix}]$

Explain why the scalar t is positive. Therefore, we have the Cholesky factorization

$A = [\begin{matrix} B & 0 \\ {\overset{鈫�}{x}}^{T} & \sqrt{t} \end{matrix}] [\begin{matrix} B^{T} & \overset{鈫�}{x} \\ 0 & \sqrt{t} \end{matrix}]$

This reasoning also shows that the Cholesky factorization of A is unique. Alternatively, you can use the LDLT factorization of A to show that (iii) implies (iv).See Exercise 5.3.63.

To show that (i) implies (ii), consider a nonzero vector, and define

role="math" localid="1659674275565" $\overset{鈫�}{y} = [\begin{matrix} \overset{鈫�}{x} \\ 0 \\ M \\ 0 \end{matrix}]$

In $R^{n}$ (fill in n 鈭� m zeros). Then

role="math" localid="1659674437541" $\overset{鈫�}{x} A^{(m)} \overset{鈫�}{x} = {\overset{鈫�}{y}}^{T} A \overset{鈫�}{y} > 0$

Sketch the curves defined in Exercises 15 through 20. In each case, draw and label the principal axes, label the intercepts of the curve with the principal axes, and give the formula of the curve in the coordinate system defined by the principal axes.

16. $x_{1} x_{2} = 1$

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