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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: Q1E (page 411)

Find the singular values of $A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}]$ .

Short Answer

Expert verified

The singular values of A are $蟽_{1} = 2$ and $蟽_{2} = 1$

Step by step solution

01

of 2: Given information

It is given that $A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}]$

By computing the eigenvalues of the square matrix, the singular values of A are found

$A^{t} A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 4 \end{matrix}]$

Then, $(x |_{2} - A^{1} A) = [\begin{matrix} x - 1 & 0 \\ 0 & x - 4 \end{matrix}]$

02

of 2: Find the singular value

Let the characteristic of polynomial $A^{1} A$ be $p (x)$

Then,

$p (x) = d e t (x I_{2} - A^{1} A) = (x - 1) (x - 4) - 0 = (x - 1) (x - 4)$

Therefore, the characteristic polynomial is $(x - 1) (x - 4)$ , which implies that the roots of $p (x)$ are 1,4 .

So, the eigenvalues of $A^{t} A$ are $位_{1} = 4$ and $位_{2} = 1$

Thus the singular values of A are $蟽_{1} = \sqrt{4} = 2$ and $蟽_{2} = \sqrt{1} = 1$

Result: $蟽_{1} = 2$ and $蟽_{2} = 1$ are the singular values of A.

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Most popular questions from this chapter

If A is a symmetric n x n matrix, what is the relationship between the eigenvalues of A and the singular values of A?

51. IfAis a symmetric $2 脳 2$ matrix with eigenvalues 1 and 2, then the angle between $\overset{鈫�}{x}$ and $A \overset{鈫�}{x}$ must be less than $蟺 / 6$ , for all nonzero vectors $\overset{鈫�}{x}$ in $R^{2}$ .

Consider a linear transformation Tfrom ${鈩�}^{m} {鈩�}^{m}$ to ${鈩�}^{n} {鈩�}^{n}$ , where $m 鈮� n$ . Show that there exist an orthonormal basis ${\overset{鈫�}{v}}_{1}, . . ., {\overset{鈫�}{v}}_{m}$ of ${鈩�}^{m} {鈩�}^{m}$ and an orthonormal basis ${\overset{鈫�}{w}}_{1}, . . ., {\overset{鈫�}{w}}_{n}$ of ${鈩�}^{n}$ such that $T ({\overset{鈫�}{v}}_{i})$ is a scalar multiple of ${\overset{鈫�}{w}}_{i}$ , for i = 1,....m.

Hint: Exercise 19is helpful.

IfA is any symmetric $2 脳 2$ matrix with eigenvalues -2 and 3, and $\overset{鈫�}{u}$ is a unit vector in ${鈩�}^{2}, {鈩�}^{2}$ , what are the possible values of $|| A \overset{炉}{u} ||$ ? Explain your answer geometrically, using Example 4as a guide.

Find the dimension of the space $Q_{2}$ of all quadratic forms in two variables.

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