Q20E If a 2 脳 2 matrix A has two dis... [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: Q20E (page 336)

If a 2 脳 2 matrix A has two distinct eigenvalues $位_{1}$ and $位_{2}$ , show that A is diagonalizable.

Short Answer

Expert verified

c $T : R^{2} 鈫� R^{2}, T x = A x$ , will be diagonal matrix. This is a diagonalization of A, so A is diagonalizable.

Step by step solution

Eigenvalues

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled.

Proving of If an n × n matrix A has n eigenvalues λ1, . . . , λn, listed with their algebraic multiplicities, then tr A = λ1 +…+λn

Let 位₁ and 位₂ be the eigenvalues of $v_{1}$ and $v_{2}$ their corresponding eigenvectors. This means at {v₁,v₂} is an eigenbasis for .In this very basis, the matrix of the linear transformation $T : R^{2} 鈫� R^{2}, T x = A x$ , will be diagonal matrix. This is a diagonalization of A, so A is diagonalizable.

Hence, A is diagonalizable.

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91影视

If a 2 脳 2 matrix A has two distinct eigenvalues $位_{1}$ and $位_{2}$ , show that A is diagonalizable.

Short Answer

Step by step solution

Eigenvalues

Proving of If an n × n matrix A has n eigenvalues λ1, . . . , λn, listed with their algebraic multiplicities, then tr A = λ1 +…+λn

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91影视

If a 2 脳 2 matrix A has two distinct eigenvalues位1and位2, show that A is diagonalizable.

Short Answer

Step by step solution

Eigenvalues

Proving of If an n × n matrix A has n eigenvalues λ1, . . . , λn, listed with their algebraic multiplicities, then tr A = λ1 +…+λn

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Logic and Functions

Applied Mathematics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

If a 2 脳 2 matrix A has two distinct eigenvalues $位_{1}$ and $位_{2}$ , show that A is diagonalizable.