The density of diagonalizable matrices of $M_n(\mathbb{C})$ problem.

Question

For any matrix $A = (a_{ij})_{1\leq i,j\leq n} \in M_n(\mathbb{C})$, we pose $||A|| = \max_{1\leq i,j\leq n} |a_{ij}|$.

$1.$ Show that $||.||$ define a norm on $M_n(\mathbb{C})$ and that $\forall A, B \in M_n(\mathbb{C})$ $$||AB||\leq n||A||||B||.$$

$2.$ Let $A_0 \in M_n(\mathbb{C})$ and $\epsilon >0$. we wanna show that: $\mathbb{B_0}(A_0,\epsilon) = \{A \in M_n(\mathbb{C})/ ||A - A_0|| < \epsilon \}$ include a diagonalizable matrix.

$(a)$ Stat that $\exists P\in GL_n(\mathbb{C})$ such that $P^{-1}A_0P = (\alpha_{ij})_{1\leq i,j\leq n}$ is upper triangular.

$(b)$ Show that the map $\Phi$ from $M_n(\mathbb{C})$ to itself with $\Phi (P) = P^{-1}AP$ is continous. $\eta \in \mathbb{R}_+^*$ such that $||A-P^{-1}A_0P||<\eta \Rightarrow||P^{-1}AP-A_0||< \epsilon$

$(c)$ Let $\lambda_1,...,\lambda_n \in \mathbb{C}$ pairwise distinct such that $|\lambda_i-\alpha_{ii}|<\eta$ for $i = 1,..,n$. We mean by $\widehat{A_0}$ the upper triangular matrix deduced from $P^{-1}A_0P$ by replacing the diagonal terms by $\lambda_i$.

Verify that $||\widehat{A_0}-P^{-1}A_0P||<\eta$ and show that $\widehat{A_0}$ and $P^{-1}\widehat{A_0}P$ are diagonalizables.

$(d)$ Deduce from what precedes that the set $D_n(\mathbb{C})$ of the diagonalizables matrices of $M_n(\mathbb{C})$ is dense in $M_n(\mathbb{C})$. What is the interior of $D_n(\mathbb{C})$.

My work:

For $1.$ I did it. and (a). But the other questions I didn't find the answer. Please help me with this problem.

Well it is ultimately about the same question: proving that the set of diagonalisable complex matrices is dense in the set of all matrices. There are more detailed indications here, but that is just one way to prove the result. Although I should say that your part (d) is really a different and more difficult question. — Marc van Leeuwen, Jan 18 '14 at 22:27

score 1 · Answer 1 · answered Jan 18 '14 at 21:58

For continuity, show that the map $P\mapsto AP$ is continuous, then that the map $P\mapsto PB$ is continuous for any $B\in\mathcal M_n(\mathbb C)$.

For question (c), notice that the extra diagonal elements of $\widehat{A_0}-P^{-1}A_0P$ are all $0$, and by construction, the modulus of the diagonal elements is smaller than $\eta$.

Notice that $\widehat{A_0}$ has distinct eigenvalues. If $M$ is a diagonalizable matrix in $\mathbb C$, so is $P^{-1}MP$ for any invertible $P$.

The density of diagonalizable matrices of $M_n(\mathbb{C})$ problem.

1 Answers1