# A few questions on matrix diagonalizability

I have a few questions on matrix diagonalizability. Suppose that $$A \in \mathbb{R}^{n \times n}$$.

1. If $$A$$ is diagonalizable, we can find $$P \in \mathbb{R}^{n \times n}$$ such that $$A = P\Lambda P^{-1}$$. Does the definition of diagonalizability in $$\mathbb{R}^{n \times n}$$ mean that eigenvalues must be real as well? If we are allowed to have complex eigenvalues, can we write more matrices $$A = P\Lambda P^{-1}$$?

2. From the Wikipedia page, I know that the set of diagonalizable matrices on $$\mathbb{C}^{n \times n}$$ is dense. If $$A \in \mathbb{R}^{n \times n}$$ is non-diagonalizable, it can be approximated with a diagonalizable matrix $$\widetilde{A} \in \mathbb{C}^{n \times n}$$ arbitrarily closely. In general $$\widetilde{A}$$ may have complex eigenvectors. Is it possible to approximate $$A$$ arbitrarily closely with real eigenvectors and complex eigenvalues?

• 1. A real matrix need not have real eigenvalues to be classified as diagonalizable. Consider e.g. pure rotation matrices. Nov 1, 2019 at 1:13
• Generally speaking, “diagonalizable” with no further qualification means diagonalizable over the same scalar field as the matrix.
– amd
Nov 1, 2019 at 6:14

1. Continuity of the eigenvalues.

Let $$A\in M_n(\mathbb{C})$$ with $$spectrum(A)=(\lambda_i)_i$$. For every $$\epsilon >0$$, there is $$\alpha >0$$ s.t. $$||A-X||<\alpha$$ implies that there exits a numbering $$(\mu_i)_i$$ of the eigenvalues of $$X$$ s.t. $$\sum_i|\lambda_i-\mu_i|<\epsilon$$.

1. One says that $$A\in M_n(K)$$ is diagonalizable over $$K$$ IFF there is $$P\in GL_n(K)$$ s.t. $$D=P^{-1}AP$$ is diagonal. That implies that the eigenvalues of $$A$$ are all in $$K$$. For example, $$\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$ is not diagonalizable over $$\mathbb{R}$$ but it is over $$\mathbb{C}$$.

2. Let $$L$$ be an extension of the field $$K$$. Note that if $$A\in M_n(K)$$ is diagonalizable over $$L$$, into the diagonal matrix $$D\in M_n(K)$$, then $$A$$ is diagonalizable over $$K$$, into the same matrix $$D$$.

3. Consider a matrix $$B\in M_n(\mathbb{R})$$ in a neighborhood of $$A\in M_n(\mathbb{R})$$. According to 1., two conjugate eigenvalues of $$A$$ give birth to $$2$$ conjugate eigenvalues of $$B$$, a simple real eigenvalue of $$A$$ to a simple real eigenvalue of $$B$$; the case of multiple real eigenvalues of $$A$$ is more complicated.

4. In any real neighborhood of $$A\in M_n(\mathbb{R})$$, there is a matrix with distinct complex eigenvalues, then diagonalizable over $$\mathbb{C}$$ but not necessarily over $$\mathbb{R}$$.

5. Let $$A=[a_{i,j}]\in M_n(\mathbb{R})$$ be a random matrix (use a normal law for each $$a_{i,j}$$). Then the eigenvalues are distinct with probability $$1$$. Yet, when $$n$$ is a large number, this matrix is "always" not diagonalizable over $$\mathbb{R}$$ because the mean of the number of real roots of a real polynomial of degree $$n$$ is in $$O(\sqrt{n})$$.