Step 1

Let, the rank of (A−3I) is 1. A matrix is diagonalizable if for each eigen value \(\displaystyle\lambda\), the rank r of the matrix

(A−3I)=n-the multiplicity of \(\displaystyle\lambda\) , where n= size of the matrix

Step 2

A \(\displaystyle{n}\times{n}\) matrix A is diagonalizable if it is similar to a diagonal matrix, that is, if there exists an invertible \(\displaystyle{n}\times{n}\) matrix c and a diagonal matrix D such that \(\displaystyle{A}={C}{D}{C}^{{−{1}}}\)

Example:

\(\displaystyle{A}={\left[\begin{array}{ccc} {1}&{0}&{0}\\{0}&{5}&{0}\\{0}&{0}&{6}\end{array}\right]}={I}_{{3}}{\left[\begin{array}{ccc} {1}&{0}&{0}\\{0}&{5}&{0}\\{0}&{0}&{6}\end{array}\right]}{{I}_{{3}}^{{-{{1}}}}}\)

Ay diagonal matrix A is diagonalizable as it is similar to itself.

Step 3

If the matrix is not diagonalizable, the one has to find a matrix with same properties consisting of eigen values on the leading diagonal and either ones or zeros on the super diagonal.

Let, the rank of (A−3I) is 1. A matrix is diagonalizable if for each eigen value \(\displaystyle\lambda\), the rank r of the matrix

(A−3I)=n-the multiplicity of \(\displaystyle\lambda\) , where n= size of the matrix

Step 2

A \(\displaystyle{n}\times{n}\) matrix A is diagonalizable if it is similar to a diagonal matrix, that is, if there exists an invertible \(\displaystyle{n}\times{n}\) matrix c and a diagonal matrix D such that \(\displaystyle{A}={C}{D}{C}^{{−{1}}}\)

Example:

\(\displaystyle{A}={\left[\begin{array}{ccc} {1}&{0}&{0}\\{0}&{5}&{0}\\{0}&{0}&{6}\end{array}\right]}={I}_{{3}}{\left[\begin{array}{ccc} {1}&{0}&{0}\\{0}&{5}&{0}\\{0}&{0}&{6}\end{array}\right]}{{I}_{{3}}^{{-{{1}}}}}\)

Ay diagonal matrix A is diagonalizable as it is similar to itself.

Step 3

If the matrix is not diagonalizable, the one has to find a matrix with same properties consisting of eigen values on the leading diagonal and either ones or zeros on the super diagonal.