---
type: "problem"
title: "A diagonalizable matrix ?"
slug: "a-diagonalizable-matrix"
author: "ancient-tree"
tags: []
difficulty: 9
qualityStatus: "unreviewed"
listed: true
origin: "Unknown"
originChapter: ""
originPage: ""
originNote: ""
license: "CC BY-SA 4.0"
---

Is the matrix $$\left(\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right)$$ diagonalizable ?

## Proof 1

_By @ancient-tree_

The only eigenvalue of this matrix is 1 (with algebraic multiplicity 2). So if it was diagonalizable, it would be similar to the identity matrix : $$\left(\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right)=P\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right) P^{-1}=\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)$$
which is a contradiction.