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Chapter 7: Q49E (page 325)

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

Short Answer

Expert verified

The required example is A=0100

Step by step solution

01

Definition of diagonalizable

The matrix A is diagonalizable if there exists an eigenbasis for A . The v鈫�1,...,v鈫�n is an eigenbasis for A , with Av鈫�1=位1v鈫�1,...,Av鈫�nV鈫�n, then the matrices

S=||||v鈫�1v鈫�2v鈫�n|||| and B=位1o鈥�00位2鈥�0鈰�鈰�鈰�鈰�00鈥�位n

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

02

Finding the suitable example

For example,

A=0100

It鈥檚 only eigenvalue is 位=0,with its corresponding eigenvectors being v=[10] .

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

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[-3040-10-273]
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