My question is $M:\mathbb{R}^{5\times 5} T:M \to M$ denote the operator $$T (A)=\frac12(A-A^T)$$ where superscript $T$ is the matrix transpose. How can I find kernel,range,nullity,rank,reel eigenvalues and eigenvectors?
Firstly, I checked T preserves sums and multiplication by scalars than I found this is linear map. $T(A+B)=T(A)+T(B) T(c.A)=c.T(A)$. If this matrix was $2\times 2$ I draw matrix $A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$ Then I could calculate $$T(A)=\frac12(A-A^T)$$ then could find kernel,range,nullity,rank.
But this matrix is $$(5\times 5)$$. So I can't draw like $2\times 2$ matrix and couldnt calculate $$T (A)=\frac12(A-A^T)$$ Can you suggest another way for me to calculate these values(kernel,range,nullity,rank,real eigenvalues and eigenvectors)?