<s>[INST] Eigenvalues and Eigenvectors of Sum of Symmetric Matrix Question:
Let A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}
Find all eigenvalues and eigenvectors of the martrix:
$$\sum_{n=1}^{100} A^n = A^{100} +A^{99} +...+A^2+A$$
I know that the eigenvectors of A are \begin{bmatrix} 1 \\ 1 \end{bmatrix} and \begin{bmatrix} 1 \\ -1 \end{bmatrix}
But I do not see any sort of correlation with the sum term and A's eigenvectors.
 [/INST] Hint: If $$A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}$$then we have $$A^2 = \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}=\begin{bmatrix} 2 & 2 \\ 2 & 2 \\ \end{bmatrix}\\A^3=\begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}\begin{bmatrix} 2&2 \\ 2&2 \\ \end{bmatrix}=\begin{bmatrix} 4&4 \\ 4&4 \\ \end{bmatrix}\\A^4=\begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}\begin{bmatrix} 4&4 \\ 4&4 \\ \end{bmatrix}=\begin{bmatrix}8&8 \\ 8&8 \\ \end{bmatrix}\\.\\.\\.\\.$$and you can prove by induction that $$A^k=\begin{bmatrix} 2^{k-1}&2^{k-1} \\ 2^{k-1}&2^{k-1}\\ \end{bmatrix}$$can you finish now?
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