When is this matrix diagonalizable?

Question

Given the matrix $$A = \left[ \begin{matrix} 2 & 2 & h & 6 \\ 0 & 4 & 2 & -2 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 4 \\ \end{matrix} \right] $$ find the value of $h$ so that the matrix is diagonalizable. $$$$ The eigenvalues of $A$ are $2,2,4,4$. Now I know $\Lambda$, but have difficulty finding a proper $P$.

Answer 1

Test for diagonalization : Let $A$ be a linear operator on an $n$ dimensional vector space $V$. Then $A$ is diagonalizable $\iff$ both of the following holds:

• $\rho_A(x)$ splits
• For each eigenvalue $\lambda$ of $A$, $$\text{multiplicity of $\lambda$ = $n-\text{rank} (A-\lambda I)$}$$

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Here, your case the first one holds.

So the question becomes, for what value of $h$, the matrices $$A-2I=\begin{pmatrix} 0 &2 &h&6\\0&2&2&-2\\0&0&0&2\\0&0&0&2 \end{pmatrix} \& \;A-4I=\begin{pmatrix} -2 &2 &h&6\\0&0&2&-2\\0&0&-2&2\\0&0&0&0 \end{pmatrix} $$

have simultaneously rank $2$ ?

Ans: $h=2$