Q73E Prove the Cayley鈥揌amilton theo... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q73E (page 360)

Prove the Cayley鈥揌amilton theorem, $f_{a} (A) = 0$ , for diagonalizable matrices A. See Exercise 7.3.54.

Short Answer

Expert verified

It is proved that the diagonalizable matrices A is $f_{A} (A) = (A - 位_{1} l) . . . . . (A - 位_{m} l) g (A) = 0$

Step by step solution

Definition

A scalar associated with a given linear transformation of a vector space and having the property that there is some nonzero vector which when multiplied by the scalar is equal to the vector obtained by letting the transformation operate on the vector especially: a root of the characteristic equation of a matrix.

Cayley鈥揌amilton theorem states that a matrix satisfies its own characteristic equation. That is, the characteristic equation of a n 脳 n matrix A is $位^{n}$ .

Diagonalizable matrices

Letbe a diagonalizable $n 脳 n$ matrices with $m 鈮� n$ distinct eigen values $位_{1}, 鈥� 位_{n}$ . Now, $f_{A} (位) = (位 - 位_{1}) . 鈥� . (位 - 位_{m}) g (位)$ where is a polynomial of degree n-m. Since is diagonaliziable we have, $(A - 位_{1} l) . 鈥� . (A - 位_{m} l) = 0$ therefore $f_{A} (A) = (A - 位_{1}) . 鈥� . (A - 位_{m} l) q (A) = 0$ .

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91影视

Prove the Cayley鈥揌amilton theorem, $f_{a} (A) = 0$ , for diagonalizable matrices A. See Exercise 7.3.54.

Short Answer

Step by step solution

Definition

Diagonalizable matrices

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91影视

Prove the Cayley鈥揌amilton theorem,fa(A)=0, for diagonalizable matrices A. See Exercise 7.3.54.

Short Answer

Step by step solution

Definition

Diagonalizable matrices

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Logic and Functions

Discrete Mathematics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Prove the Cayley鈥揌amilton theorem, $f_{a} (A) = 0$ , for diagonalizable matrices A. See Exercise 7.3.54.