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I would like an example of an anti symmetric matrix for which the $\lambda$-eigenspace is a strict subset of the generalized $\lambda$-eigenspace. If no such anti symmetric matrix exists I would like a proof of this. All of this is over the field C. By transpose I mean the following:

if $A^T=B$ then $A_{ij} = B_{ji}$

Mathew
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  • You need to do a little work. The reason that a symmetric (Hermitian) matrix $A$ is diagonalisable is because if $S$ is $A$ invariant then so is $S^\bot$ is $A$ invariant. Perhaps you can see if the same can be used for anti symmetric case? (I am assuming that this is over $\mathbb{C}$.) – copper.hat Jan 07 '20 at 17:32
  • yes, I am over C – Mathew Jan 07 '20 at 17:35
  • To clarify: You're working over $\Bbb C$. By symmetric (anti-symmetric), you mean hermitian (skew-hermitian) or are you talking just about the ordinary transpose? – Ted Shifrin Jan 07 '20 at 17:45
  • I mean transpose in the sense of swapping entries without conjugation – Mathew Jan 07 '20 at 17:46
  • According to problem 34 of section 4.4 of Horn and Johnson's matrix analysis, such a matrix exists and the smallest example has size $3$. – Ben Grossmann Jan 07 '20 at 20:00

1 Answers1

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Note that the matrix $$ A = \pmatrix{1&i\\i&-1} $$ is symmetric but fails to be diagonalizable (since $A^2 = 0$). It follows that $$ M = \pmatrix{0&1\\-1&0} \otimes A = \pmatrix{0&0&1&i\\0&0&i&-1\\-1&-i&0&0\\-i&1&0&0} $$ is skew-symmetric and also fails to be diagonalizable.

Notable facts, Horn and Johnson's Matrix Analysis, second edition.

Corollary 4.4.19: Let $A \in M_n$ be skew-symmetric. Then $r = \operatorname{rank}(A)$ is even, the non-zero singular values of $A$ occurs in pairs $\sigma_1 = \sigma_2 = s_1 \geq \sigma_3 = \sigma_4 = s_2 \geq \cdots \geq \sigma_{r-1} = \sigma_r = s_{r/2} \geq 0$, and $A$ is unitarily congruent to $$ 0_{n-r} \oplus \pmatrix{0&s_1\\-s_1 & 0} \oplus \cdots \oplus \pmatrix{0&s_{r/2}\\-s_{r/2} & 0}. $$

and fact 2:

4.4.P34: Although a symmetric complex matrix can have any given Jordan canonical form (4.4.24), the Jordan canonical form of a skew-symmetric complex matrix has a special form. It consists of only the following three types of direct summands: (a) pairs of the form $J_k(\lambda) \oplus J_k(-\lambda)$, in which $\lambda \neq 0$; (b) pairs of the form $J_k(0) \oplus J_k(0)$, in which k is even; and (c) $J_k(0)$, in which k is odd.

Ben Grossmann
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