Given that B is a symmetric matrix how can I show that if B can be diagonalized then there exists an orthonormal basis of eigenvectors of B?

Asked Oct 07 '19 at 14:28

Active Oct 07 '19 at 15:01

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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.

edited Oct 07 '19 at 15:01

asked Oct 07 '19 at 14:28

Eisen

2

As a response to your edit: What do you understand by a basis? – Sten Oct 07 '19 at 15:10
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. – Arthur Oct 07 '19 at 18:55

Given that B is a symmetric matrix how can I show that if B can be diagonalized then there exists an orthonormal basis of eigenvectors of B?

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