= Solution
Regard the real <skew-symmetric matrix> as acting on $\mathbb C^n$. Its <Hermitian adjoint> is $A^*=A^T=-A$. Parts (i) and (iii) show that its <eigenvalues> are purely imaginary and its complex <minimal polynomial> has no repeated roots.
Since $A$ has real entries, nonreal <eigenvalues> occur in conjugate pairs. A nonzero pair has the form $i\alpha,-i\alpha$, with $\alpha>0$, and contributes the real <quadratic polynomial>
$$
(x-i\alpha)(x+i\alpha)=x^2+\alpha^2.
$$
Zero, if present, contributes one factor $x$. Hence the real <minimal polynomial> is
$$
\boxed{m_A(x)=x^{\varepsilon}\prod_{j=1}^r(x^2+\alpha_j^2),\qquad
\varepsilon\in\{0,1\},\quad\alpha_j>0\text{ distinct}.}
$$
The product annihilates $A$ after <complexification of a real vector space>, so it also annihilates the original real <linear map>; no factor can be removed because each displayed complex root is an <eigenvalue>. This proves both the asserted form and its minimality. The case $A=0$ has $\varepsilon=1$ and an empty product.
Back to article page
