# How to prove real symmetric matrix is diagonalizable (with reference request)

I want to know a general linear algebra theories that are needed in proving real symmetric matrix is diagonalizable.

I know of eigenvalues, eigenvectors, eigenspaces in the context of diagonalizability. But I haven't read about Gram–Schmidt orthogonalization and its related concepts yet.

What area of linear algebra theory should I understand in general to deal with and to prove statements like the above?

I have Friedberg's Linear Algebra text. If you have read it, you may give me the pages to read.

And if not long, give me some scratch of the proof of the statement.

Thanks.

The proof I have seen first proves that any square matrix $A$ can be written as $A = Q U Q^{-1}$, where $Q$ is unitary, and $U$ is upper triangular. The proof uses the fact that any matrix has at least one eigenvalue, it uses the Gram-Schmidt process, and it proceeds by induction. It is called a Schur decomposition.
Then you argue that if A is real symmetric, then U must be real diagonal, since $A^* = Q U^* Q^{-1}$ where the star denotes the conjugate transpose.
• But $Q$ is a complex matrix. Can we assure that $Q$ is real? – awllower Dec 13 '17 at 9:04
• You go through the argument of Schur's decomposition. It is by induction, and starts by choosing the first column of $Q$ as an eigenvector. If $A$ is real symmetric, then the eigenvector can be chosen as real. – Stephen Montgomery-Smith Dec 13 '17 at 14:16