Exercises: Diagonalization of Matrices

Problem 1

For each of the following matrices \(\Mtrx{A}\) determine if it is diagonalizable. If it is, find a matrix identity \(\Mtrx{D} = \Mtrx{C}^{-1} \Mtrx{A} \Mtrx{C}\), with \(\Mtrx{D}\) a diagonal matrix.

  1. \[ \Mtrx{A} = \left[ \begin{array}{cc} 1 & 3 \\ 1 & 2 \end{array} \right] \]
  2. \[ \Mtrx{A}\ =\ \left[ \begin{array}{ccc} 2 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{array} \right] \]
  3. \[ \Mtrx{A}\ =\ \left[ \begin{array}{rrrrr} 2 & 1 & -3 & 7 & -5 \\ 0 & 4 & 4 & -1 & 4 \\ 0 & 0 & 2 & 1 & -9 \\ 0 & 0 & 0 & 2 & 3 \\ 0 & 0 & 0 & 0 & -1 \end{array} \right] \]
Problem 2

Find matrices \(\Mtrx{P}\) and \(\Mtrx{Q}\) such that \(\Mtrx{Q}\Mtrx{A}\Mtrx{P}\) is diagonal, where

\[ \Mtrx{A} = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right] \]
Problem 3

Let \(\Mtrx{D}\) be an \((n,n)\)-diagonal matrix with diagonal entries \(d_1\), ... , \(d_n\). If \(\Mtrx{X}\) is an arbitrary invertible \((n,n)\)-matrix, let

\[\Mtrx{A}\ \DefEq\ \Mtrx{X}\Mtrx{D}\Mtrx{X}^{-1}\]

  1. Is \(\Mtrx{A}\) always invertible?

  2. Is \(\Mtrx{A}\) always symmetric?

  3. Is \(\Mtrx{A}\) always diagonalizable?

Problem 4

Suppose for an \((n,n)\)-matrix \(\Mtrx{A}\) there exist invertible \((n,n)\)-matrices \(\Mtrx{X}\) and \(\Mtrx{Y}\) such that \(\Mtrx{D}\) and \(\Mtrx{E}\) defined as

\(D\)\(\DefEq\)\(\Mtrx{X} \Mtrx{A} \Mtrx{X}^{-1}\)
\(E\)\(\DefEq\)\(\Mtrx{Y} \Mtrx{A} \Mtrx{Y}^{-1}\)

are diagonal. Is always \(\Mtrx{D} = \Mtrx{E}\)?

Problem 5

Let \(\Mtrx{A}\) be an \((n,n)\)-diagonalizable matrix with characteristic polynomial

\[p(\lambda) = a_n\lambda^n + \cdots + a_1\lambda + a_0\]

Let \(p(\Mtrx{A}) \DefEq a_n A^n + \cdots + a_1 A + a_0 \IdMtrx{n}\). – Show that \(p(\Mtrx{A}) = 0\).