grandes-ecoles 2013 Q18

grandes-ecoles · France · x-ens-maths__pc Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$, and that this sequence converges to a diagonal matrix $D$.
Finally, show that the diagonal entries of $D$ are the eigenvalues of $A^{(0)}$. What can be said about their multiplicities?

Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$, and that this sequence converges to a diagonal matrix $D$.

Finally, show that the diagonal entries of $D$ are the eigenvalues of $A^{(0)}$. What can be said about their multiplicities?

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