3
$\begingroup$

Let $A\in M_{n}(C)$ be a matrix such that $A^*=-A$ and $A^4=I$.

I need to prove that the eigenvalues of A are between $-i$ to $i$ and that $A^2+I=0$

I didn't get to any smart conclusion.

Thanks

  • 1
    Hint for the first part: if z is an eigenvalue of A, show that z^4=1.2011-08-11
  • 4
    Since $A^*=-A$, i.e. $A$ is skew-hermitian, you can diagonalize. Try that and see what you get.2011-08-11
  • 4
    What does "between $−i$ to $i$" mean?2011-08-11
  • 1
    "Between" may be translating, from a language different from English, words that mean "in" or "among" [the two elements of a 2-element set].2011-08-12
  • 0
    Can't you say in english "all the numbers between this value to another". I believe you can..2011-08-12
  • 0
    @John M :Can you extend your hint please? I didn't understand much.2011-08-12
  • 1
    One can (provided one uses *between this and that* instead of *between this to that*), but only for numbers in a totally ordered set such as the real line, and hardly for the complex plane. Or you wish to speak of the set $\{t\mathrm{i}\mid t\ \text{real}, -1\le t\le 1\}$, then I believe you can call it the **segment** between $\mathrm{-i}$ and $\mathrm{i}$.2011-08-12

1 Answers 1

1

Do you recall that hermitian matrices ($A^*=A$) must have real eigenvalues? Similiarly, skew-hermitian matrices, i.e. $A^*=-A$, must have pure imaginary eigenvalues. (see Why are all nonzero eigenvalues of the skew-symmetric matrices pure imaginary?)

Also, since $A$ is skew-hermitian, then $A$ is normal too, i.e. $A^*A=AA^*$, so we can apply the spectral theorem: there exists a unitary matrix $U$ such that $A=UDU^{-1}$, where $D$ is a diagonal matrix, whose diagonal entries are the eigenvalues of $A$.

Thus we know that $A^4=(UDU^{-1})^4=UD^4U^{-1}=I$, so $D^4=I$, so all the eigenvalues are 4th roots of unity, i.e. $1,-1,i,\text{ or} -i$. But we already know the eigenvalues are pure imaginary, so all the eigenvalues are $i$ or $-i$. So $D^2=-I$.

Finally, we have $A^2=(UDU^{-1})^2=UD^2U^{-1}=U(-I)U^{-1}=-I$, i.e. $A^2+I=0$.

  • 0
    This theorem can be applied only when A is normal? Is this the only mandatory condition?2011-08-12
  • 0
    Yes, in $M_n(\mathbb{C})$, $A$ diagonalizes in a orthonormal basis $\Leftrightarrow$ $A$ is normal.2011-08-12