Linear Algebra Help needed symmetric matrices!

[LoveMjj]

1. Show that for all n X n symmetric matrices A, the matrix \(\displaystyle A^{2} + I_{n}\) is invertible.



2. Find an orthonormal eigenbasis for each symmetric matrix A


a) \(\displaystyle A=\begin{bmatrix}3 & 2\\2 & 1\end{bmatrix}\)


b) \(\displaystyle A= \begin{bmatrix}0 & 0& 1\\ 0 & 2 & 0\\ 1 & 0 & 0\end{bmatrix}\)



C) \(\displaystyle A= \begin{bmatrix}1& 0& 0& 1\\ 0&0&0&0\\0&0&0&0\\ 1&0&0&1\end{bmatrix}\)



can someone please tell me how to solve these in the most simplest terms possible? Any help would be GREATLY appreciated!

 

[galactus]

To find the orthonormal eigenbases for a matrix A, find the eigenvalues.

Then, for each one of them, find a basis for the kernel \(\displaystyle Ker (cI-A)\), where c is an eigenvalue.

Then, use the Gram-Schmidt process to convert to an orthonormal basis.

For instance, for the first one, the characteristic polynomial is \(\displaystyle \lambda ^{2}-4\lambda -1\)

The roots, and thus the eigenvalues, are \(\displaystyle \lambda = -(\sqrt{5}-2), \;\ \lambda = \sqrt{5}+2\)

\(\displaystyle (\sqrt{5}+2)\cdot I_{2}-A=\begin{bmatrix} \sqrt{5}-1 & -2\\ -2 & \sqrt{5}+1\end{bmatrix}\)

Now, find the kernel and use GS to find the orthonormal basis.