Short Answer

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

The required example is $A = (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix})$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of diagonalizable

The matrix A is diagonalizable if there exists an eigenbasis for A . The ${\vec{v}}_{1}, . . ., {\vec{v}}_{n}$ is an eigenbasis for A , with $A {\vec{v}}_{1} = λ_{1} {\vec{v}}_{1}, . . ., A {\vec{v}}_{n} {\vec{V}}_{n}$ , then the matrices

$S = [\begin{matrix} | & | & | | \\ {\vec{v}}_{1} & {\vec{v}}_{2} & {\vec{v}}_{n} \\ | & | & | | \end{matrix}]$ and $B = [\begin{matrix} λ_{1} & o & \dots & 0 \\ 0 & λ_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & λ_{n} \end{matrix}]$

Will be diagonalize A , meaning that $S^{- 1} A S = B$ .

Step 2: Finding the suitable example

For example,

$A = (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix})$

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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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Definition of diagonalizable

Step 2: Finding the suitable example

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Definition of diagonalizable

Step 2: Finding the suitable example

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