Product of a symmetric and anti-symmetric matrix

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The matrix product does not preserve the symmetric nor the anti-symmetric property. A simple example of this phenomenon is the following.

Pick

$$S=\begin{pmatrix} 2 & 1\\ 1 &2 \end{pmatrix} \quad \text{and} \quad A= \begin{pmatrix} 0 & -1\\ 1 &0 \end{pmatrix}.$$

Then $$SA=\begin{pmatrix} 2 & 1\\ 1 &2 \end{pmatrix} \begin{pmatrix} 0 & -1\\ 1 &0 \end{pmatrix}= \begin{pmatrix} 1 & -2\\ 2 &-1 \end{pmatrix}$$

which is symmetric nor anti-symmetric. Similarly,

$$AS= \begin{pmatrix} 0 & -1\\ 1 &0 \end{pmatrix}\begin{pmatrix} 2 & 1\\ 1 &2 \end{pmatrix}= \begin{pmatrix} -1 & -2\\ 2 &1 \end{pmatrix}$$

Again, this is not symmetric nor anti-symmetric.

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Student W
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Student W

Applied mathematics student. Interests: Analysis, Particle systems, Stochastics

Updated on August 01, 2022

Comments

  • Student W
    Student W over 1 year

    I have the following question about matrices, Let $S$ and $A$ be two $n \times n$ matrices which are respectively symmetric and anti-symmetric. Can I conclude anything about the products $SA$ or $AS$, are they symmetric or anti-symmetric?

    This is part of a bigger problem where I have already shown, $$ \langle x, Ax \rangle = 0$$ For $A$ antisymmetric, but I require that $$ \langle Mx, Ax \rangle = 0$$ For some matrix M, what conditions could I impose on $M$ to satisfy this, I was hoping symmetry would be sufficient or do I require something stronger such as diagonality?