Diagonalization of symmetric matrices

Question

Mas-Collel, Whinston and Green's Microeconomic Theory (3rd edition) asserts that any symmetric matrix can be diagonalized as following:

What would be the proof of this?

Answer 1

If $M$ is a symmetric matrix then we can choose eigenvectors $v_1,\ldots,v_N$ of $M$ such that $B:=\{v_1, \ldots, v_N\}$ forms an orthonomal basis for $\mathbb{R}^N$. Let $C$ be be the change of basis matrix which takes the standard orthonomal basis of $\mathbb{R}^N$ to $B$. The matrix $C$ has columns given by the eigenvectors $v_i$, that is $$C = \begin{pmatrix} v_1 & v_2 & \cdots & v_N\end{pmatrix}.$$

Since the columns of $C$ are an orthonomal basis, $C$ is invertible with $C^{-1} = C^T$. The desired diagonal matrix is then given by $D = C^{-1} M C$.

Intuitively, what we have done is used $C$ to transform the standard basis to a basis which behaves nicely with respect to $M$ (namely the eigenbasis $B$). Since $M$ acts on eigenvectors by scaling, we see that $M$ behaves as though it is diagonal with respect to this basis. We then transform back to the standard basis using $C^{-1}$.

Nb: I may have mixed up which $C$ has the inverse.