I've found that the eigenvalues are 2 and 1 but how do I find a so that C is diagonalizable?
2026-03-30 21:46:10.1774907170
Linear Algebra: Diagonalization (finding a so that matrix is diagonalizable)
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The characteristic polynomial is $$(\lambda-1)^2(\lambda-2)$$
The algebraic multiplicity for $1$ is $2$.
Look at matrix $$C-I=\begin{bmatrix} 1 & 0 & 1 \\ 1 & 0 & a \\ 0 & 0 & 0\end{bmatrix}$$
We need the nullity of $C-I$ , which is also the dimension of eigenspace corresponding to eigenvalue $1$, to be $2$ in order to diagonalize it. The two non-zero rows should be multiple of each other.