I have these general wondering about matrices but I don't know to proceed with a proof or a counter example. Suppose that $A$ (dimension $n\times n$) is a real symmetric matrix.
- If $A$ has $n$ eigenvalues that are all $1$'s, does $A$ equal the identity matrix?
- If $A$ has $n$ eigenvalues that are all $0$'s, does $A$ equal the zero matrix?
Can someone elucidate things for me please?
Edit: I learned/can look up diagonalization theorems for real matrices.