Determine whether B = [2 i i 4] is diagonalizable over C.

Question

Determine whether [latex]B = \left [ \begin{matrix} 2 & i \ i & 4 \end{matrix} ight ][/latex] is diagonalizable over [latex]\mathbb{C}[/latex].

Accepted Answer

We have

[latex]C(\lambda) = det(B − \lambda I) = \left [ \begin{matrix} 2 - \lambda & i \ i & 4 - \lambda \end{matrix} ight ] = \lambda ^2 − 6\lambda + 9 = (\lambda − 3)^2[/latex]

So, the only distinct eigenvalue of B is [latex]\lambda_1 = 3[/latex] with algebraic multiplicity 2.

For [latex]\lambda_1 = 3[/latex],

[latex]B − \lambda_1 I = \left [ \begin{matrix} -1 & i \ i & 1 \end{matrix} ight ] \sim \left [ \begin{matrix} 1 & -i \ 0 & 0 \end{matrix} ight ][/latex]

Hence, the geometric multiplicity of [latex]\lambda_1[/latex] is 1. Therefore, B is not diagonalizable since the geometric multiplicity of [latex]\lambda_1[/latex] is less than its algebraic multiplicity.

Question 9.4.2: Determine whether B = [2 i i 4] is diagonalizable over C.

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