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Linear Algebra With Applications
Found in: Page 325
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Give an example of a matrix A of rank 1 that fails to be diagonalizable.

The required example is A=0100

See the step by step solution

Step by Step Solution

Step 1: Definition of diagonalizable

The matrix A is diagonalizable if there exists an eigenbasis for A . The v1,...,vn is an eigenbasis for A , with Av1=λ1v1,...,AvnVn , then the matrices

S=||| |v1v2vn||| | and B=λ1o00λ2000λn

Will be diagonalize A , meaning that S-1AS=B .

Step 2: Finding the suitable example

For example,


It’s only eigenvalue is λ=0 ,with its corresponding eigenvectors being v=[1 0] .

However, this does not make for an eigenbasis, so A is not diagonalizable.

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