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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 7: Q46E (page 338)

In all the part of this problem, consider a matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ with the eigenvalues $位_{1} and 位_{2}$ .
a. show that $位_{1}^{2} + 位_{2}^{2} = a^{2} + d^{2} + 2 bc$
b. show that $位_{1}^{2} + 位_{2}^{2} 鈮� a^{2} + d^{2} + c^{2} + d^{2}$
c. for which matrices A Does the equality $位_{1}^{2} + 位_{2}^{2} = a^{2} + d^{2} + c^{2} + d^{2}$

Short Answer

Expert verified

The part of this problem, consider a matrix.

$d e t (A - 位 l) = 0 |\begin{matrix} a - 位 & b \\ c & d - 位 \end{matrix}|$

a) $= a^{2} + d^{2} + 2 b c$

b) $位_{1}^{2} + 位_{2}^{2} = a^{2} + d^{2} + 2 bc 鈮� a^{2} + b^{2} + c^{2} + d^{2}$

c) $b^{2} + c^{2} = 2 bc 鈬� b^{2} = 2 bc + c^{2} = 0 鈬� {(b - c)}^{2} = 0 鈬� b = c$

Step by step solution

01

definition of matrix

A function is defined as a relationship between a set of inputs that each have one output.

Given,

$A = [\begin{matrix} a & b \\ c & d \end{matrix}] d e t (A - 位 l) = 0 |\begin{matrix} a - 位 & b \\ c & d - 位 \end{matrix}|$

Multiplication:

$(a - 位) (d - 位) - bc = 0 位^{2} - (a + d) 位 + (ad - bc) = 0 位^{2} = a + d 卤 \frac{\sqrt{{(a + d)}^{2}} - 4 (ad - bc)}{2}$

a. Given, $位_{1}^{2} + 位_{2}^{2} = a^{2} + d^{2} + 2 bc$

we compute,

$位_{1}^{2} + 位_{2}^{2} = \frac{2 {(a + d)}^{2} + 2 {(a + d)}^{2} - 8 (ad - bc)}{4} = {(a + d)}^{2} - 2 (ad - bc) = a^{2} + d^{2} 鈮� 2 bc$

02

square the value

b. given, $位_{1}^{2} + 位_{2}^{2} 鈮� a^{2} + b^{2} + c^{2} + d^{2}$

apply the ${(b - c)}^{2} 鈮� 0, b^{2} + c^{2} - 2 bc 鈮� 0,$

$鈬� b^{2} + c^{2} 鈮� 2 b c$

Now,

using part a, we get the $位_{1}^{2} + 位_{2}^{2} = a^{2} + d^{2} + 2 bc 鈮� a^{2} + b^{2} + c^{2} + d^{2}$

c.Given, $位_{1}^{2} + 位_{2}^{2} = a^{2} + b^{2} + c^{2} + d^{2}$

The equality holds if and only

if $b^{2} + c^{2} = 2 b c 鈬� b^{2} - 2 b c + c^{2} = 0 鈬� {(b - c)}^{2} = 0 鈬� b = c$

hence,

a) $= a^{2} + b^{2} + 2 bc$

b) $位_{1}^{2} + 位_{2}^{2} = a^{2} + d^{2} + 2 bc 鈮� a^{2} + b^{2} + c^{2} + d^{2}$

c) $b^{2} + c^{2} = 2 b c 鈬� b^{2} - 2 b c + c^{2} = 0 鈬� {(b - c)}^{2} = 0 鈬� b = c$

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Most popular questions from this chapter

Is $\overset{鈬赌}{v}$ an eigenvector of $A + 2 I_{n}$ ? If so, what is the eigenvalue?

Find an eigenbasis of given matrix and diagonalize it.

$A = (\begin{matrix} 1 & 3 \\ 2 & 6 \end{matrix})$

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

$(\begin{matrix} 0 & - 1 \\ 1 & 2 \end{matrix})$

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

$[\begin{matrix} 3 & - 2 & 5 \\ 1 & 0 & 7 \\ 0 & 0 & 2 \end{matrix}]$

find an eigenbasis for the given matrice and diagonalize:

$A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{matrix}]$

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