Let $A$ be a Hermitian matrix, then which of the following statements is/are true?

Question

(a) The diagonal entries of $A$ are all real.

(b) There exist a unitary matrix $U$ such that $U^*AU$ is diagonal matrix.

(c) If $A^3$=$I$ ,then $A=I$.

(d) If $A^2$=$I$ ,then $A=I$

What I've done :

for(a); (T) since $a_{ii}=\bar{a_{ii}} \forall i$ so if we let $a_{ii}$= $x+iy$, then $a_{ii}$=$x$ which is real.

for(d); (F) since I've a counter example as $A$= $\begin{bmatrix} 0 & 1\\ 1 & 0\\ \end{bmatrix}$

But I don't have any idea about (b) & (c).

Please explain how to think...

Answer 1

Your approach for (a) and (d) is correct. Now as we have a Hermitian matrix so it will be diagonalizable and all eigenvalues will be real, by Cayley-Hamilton Theorem $A^3=I \implies \lambda^3=1$ (say $\lambda$ is an eigenvalue of $A$), you can conclude from here $A=I$.

Hint for (b): Spectral Theorem, if $A$ is Hermitian, then $A$ is unitarily diagonalizable.

Answer 2

Hint for (c): If $A^{3}=I$, then the polynomial

$$p\left(x\right)=x^{3}-1=\left(x-1\right)\left(x^{2}+x+1\right)$$

vanishes on $A$, i.e. $p\left(A\right)=0$. Notice that $b^{2}-4ac=1^{2}-4\cdot1\cdot1=-3<0$.