A
is symmetric (or selfadjoint, if your matrices are complex), so it is diagonalizable.
It is not idempotent, because A2=5A
.
It is not nilpotent, because An=5n−1A
.
The minimal polynomial of A
is pm(t)=t(t−5), while the characteristic polynomial is pc(t)=t4(t−5)
. So they are different.
The characteristic polynomial can be obtained this way: we have
A=⎡⎣⎢⎢1⋮10⋮0⋯⋱⋯0⋮0⎤⎦⎥⎥⎡⎣⎢⎢⎢⎢⎢100⋯⋯⋱⋯100⎤⎦⎥⎥⎥⎥⎥,
so its spectrum agrees with that of
⎡⎣⎢⎢⎢⎢⎢100⋯⋯⋱⋯100⎤⎦⎥⎥⎥⎥⎥⎡⎣⎢⎢1⋮10⋮0⋯⋱⋯0⋮0⎤⎦⎥⎥=⎡⎣⎢⎢⎢⎢⎢50000⋯⋯⋱⋯000⎤⎦⎥⎥⎥⎥⎥.
The eigenvalues are then 5,0,0,0,0, and so pc(t)=t4(t−5). As A5−5A=0, the minimal polynomial is at most t(t−5). But it also requires 0 and 5 as roots, so pm(t)=t(t−5).