Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Once again, can the great maths minds here please give a hand to explain or solve this problem. Much appreciated!

How can you prove that a bi-symmetric matrix multiplied by symmetrical vector will give me a symmetrical vector? Vice versa, a bi-symmetric matrix multiplied by anti-symmetrical vector will give me an anti-symmetrical vector?

Is there a way to solve it without using symbolic examples?

share|cite|improve this question
What is a symmetric vector? – Dylan Moreland Jul 4 '12 at 7:32
What I meant to say is a symmetrical vector, $v=[a_1 a_2 a_3 a_4 a_3 a_2 a_1]^T$. As for an anti-symmetrical vector $v=[a_1 a_2 a_3 a_4 -a_3 -a_2 -a_1]^T$. Not sure if that is the sensible term for it? – JuniorEngie Jul 4 '12 at 8:24
up vote 1 down vote accepted

Let $J$ be the (as wikipedia calls it) exchange matrix \[ J = \begin{pmatrix} 0 & 0 & \cdots & 1\\\ 0 & 0 & \dots & 0\\\ & & \vdots &\\\ 0 & 1 & \cdots & 0\\\ 1 & 0 & \cdots & 0 \end{pmatrix} \] I assume that you call $v \in K^n$ symmetric if $Jv = v$ and anti-symmetric if $Jv = -v$, right?

Now let $A \in \mathrm{Mat}(n, K)$ bisymmetric, that is $JA = AJ$ and $A^T = A$. If $v \in K^n$ is symmetic, then \[ J(Av) = JAv = AJv = Av \] so $Av$ is symmetric, if $Jv = -v$, then \[ J(Av) = AJv = A(-v) = -Av. \] Note, that we didn't use $A^T = A$ (i. e. $A$ being symmetric).

share|cite|improve this answer
Thank you martini! Your definition is brilliant, clean and simple! Million of thanks~ – JuniorEngie Jul 4 '12 at 8:17

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.