What is the condition for a matrix to have distinct eigenvalues.
We are working on large symmetric matrices where finding its characteristic equation is next to impossible. Still we need to check whether the eigenvalues are distinct or not. Please help, if there is any theory rtgarding it.
A theory that I use at times is the Interlacing Inequalities for symmetric matrices. If $A$ is symmetric (or Hermitian in general), with eigenvalues $$\lambda_1 \geq \lambda_2 \geq \cdots \geq\lambda_n$$ and has a principal submatrix with eigenvalues $$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_k$$ then we have the following interlacing inequality;
$$\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2\geq \cdots \geq \lambda_k \geq \mu_k \geq \lambda_{k+1}$$
This can be useful if we know that a principal submatrix has distinct eigenvalues. The interlacing allows us to split the eigenvalues of $A$.
This theorem of course has limited application in that it requires knowledge of eigenvalues for principal submatrices.