Decompose matrix A into its symmetric and skew-symmetric parts

Question

I have this square matrix:

$$A=\begin{pmatrix}0&-14\\14&0\end{pmatrix}$$

and I want to find its symmetric and skew-symmetric parts but I am confuse because it is already a skew symmetric matrix, and when finding the symmetric part I get a zero matrix. Is that possible?

Answer 1

In general, you can decompose $A$ uniquely as

$$A=\frac{A+A^{T}}{2}+\frac{A-A^{T}}{2}$$

with $A_{\rm sym}=\frac{A+A^{T}}{2}$ the symmetric part and $A_{\rm anti-sym}=\frac{A-A^{T}}{2}$ the anti-symmetric part. If $A$ is anti-symmetric then $A^{T}=-A$ and you get

$$A_{\rm sym}=0$$

$$A_{\rm anti-sym}=A$$

Thus, the symmetric part of an anti-symmetric matrix is indeed zero. There is no problem with your conclusion.