In This section we introduce a kind of must-be diagonalization, and the similarity transformation matrix may be the matrix of the orthogonal matrix, that is, the real symmetric matrix.

theorem 1 $n The eigenvalues of a solid symmetric matrix are real numbers.

Tip: $Ax =\lambda x, \bar\lambda \bar{x} = \bar{ax}=\bar{a}\bar{x}=a\bar{x}$, thus $\bar{x}^t a^{t}=\bar{x}^t A=\bar\lambda \ bar{x}^t$, so $$\lambda\bar{x}^tx=\bar{x}^t ax=\bar\lambda\bar{x}^tx,$$

Note that $\bar{x}^tx=\bar{x_1}x_1 +\cdots + \bar{x_n}x_n = |x_1|^2 + \cdots +|x_n|^2$, that is, the conclusion.

Inference 1 $n $ order real symmetric matrices must have $n$ real eigenvalues (heavy root by weight).

theorem 2 is set $\lambda_1, \lambda_2$ is a real symmetric matrix $a$ two eigenvalues, $p _1, p_2$ is the corresponding eigenvector. $\lambda_1 \neq, \lambda_2$, $p_1 orthogonal.

Tip: $ (ap_1) ^tp_2= (\lambda_1p_1) ^tp_2=\lambda_1p_1^t$, and

$ (ap_1) ^tp_2=p_1^ta^tp_2=p_1^tap_2=\lambda_2p_1^tp_2$, again by $p_1^tp_2=[p_1, p_2]$, is the conclusion.

theorem 3 set $a$ is a $n$-order symmetric matrix, there is a $n$-order orthogonal matrix $p$, which makes $p^{-1}ap = Diag (\lambda_1, \cdots, \lambda_n) $. Where $\lambda_1, \cdots, \lambda_n$ are all eigenvalues of $a$ (both real numbers). $P = (p_1, \cdots, P_n) $ corresponding canonical orthogonal eigenvectors (also real).

Tip: Mathematical induction.

Inference 2 sets $a$ as a real symmetric matrix, if $\lambda$ is a characteristic equation $| A-\lambda e|=0$ $k$ Root, the $\lambda$ corresponds to the $k$ linearly independent eigenvector.

to $a$ the real symmetric matrix diagonally, proceed as follows :

(1) The whole eigenvalue of $a$ is obtained, and the diagonal element of diagonal matrix is gotten.

(2) for the single-weight eigenvalue, the corresponding 1 linearly independent eigenvectors are calculated and the units are set. For $k$ eigenvalue, it $k$ a linearly independent eigenvector, and then, according to Schmidt method, orthogonal its specification, and obtains $k$ orthogonal unit eigenvectors. Finally, we get $n$ orthogonal eigenvector $p_1, \cdots, p_n$.

(3) Make $p= (P_1, \cdots, P_n) $, which is an orthogonal matrix and has $p^{-1}ap=p^tap=diag (\lambda_1, \cdots, \lambda_n) $.

Proposition set $ A, b$ is a $n$-order symmetric matrix, then $a$ and $b$ similar necessary and sufficient conditions for $a$ and $b$ have the same eigenvalues.

Note : If $ A, b$ is not a real symmetric matrix, then adequacy is not established. such as A=[0 0;0 0], b=[0 1;0 0], eigenvalues are all 0, but their rank is different, and thus certainly not a similarity matrix.

5.4 Diagonal of real symmetric matrices