Suppose that $B \in R^{n \times n}$ is a symmetric matrix.

How do I prove that if $B$ can be diagonalized, then there must be an orthonormal basis ($v_1,v_2, \dots, v_n$) of eigenvectors of $B$?

This is different than the question tagged as duplicate because I am asking for a proof for that there must be an orthonormal basis rather than just proving vectors are orthogonal.

  • 2
    $\begingroup$ As a response to your edit: What do you understand by a basis? $\endgroup$
    – Sten
    Oct 7, 2019 at 15:10
  • $\begingroup$ It is actually basically the same. With Gram-Schmidt you can find an orthonormal basis for each eigenspace. That's easy. The difficult part is showing that the different eigenspace are orthogonal. That's what the other post deals with. Thus the duplicate. $\endgroup$
    – Arthur
    Oct 7, 2019 at 18:55