Symmetric and anti-symmetric matrices

[Yankel]

Hello all,

I have 3 matrices, A - symmetric, B - anti symmetric, and P - any matrix

All matrices are of order nXn and are not the 0 matrix

I need to tell if the following matrices are symmetric or anti symmetric:

1) 5AB-5BA
2) 4B^3
3) A(P^t)(A^t)
4) (A+B)^2
5) BAB

How would you approach this, are there any known relations between symmetric and anti symmetric matrices ? Thanks !

 

[Fallen Angel]

Hi Yankel,

If you don't know the relations just do the computations, I mean, call
$A=(a_{i,j})_{i,j=1}^{n}$, $B=(b_{i,j})_{i,j=1}^{n}$ and $P=(p_{i,j})_{i,j=1}^{n}$.

Now, for example the entry $(i,j)$ of the matrix $AB$ is the product of the $i$-th row of $B$ times the $j$-th column of $B$ so it is
$\displaystyle\sum_{k=1}^{n}a_{i,k}b_{k,j}$

Now you have to check if this is equal, inverse of nothing to the $(j,i)$ entry of the same matrix, that will be
$\displaystyle\sum_{k=1}^{n}a_{j,k}b_{k,i}$

For doing this, take into account that $a_{i,j}=a_{j,i}$ and $b_{i,j}=-b_{i,j}$ for every $i\neq j$. (Sometimes in antisymmetric matrices $a_{i,i}=0$ but sometimes $a_{i,i}$ could be whatever, it depends on the definition of antisymmetry you are working with)

The same idea can be extended to all your cases.

 

[Opalg]

Hello all,

I have 3 matrices, A - symmetric, B - anti symmetric, and P - any matrix

All matrices are of order nXn and are not the 0 matrix

I need to tell if the following matrices are symmetric or anti symmetric:

1) 5AB-5BA
2) 4B^3
3) A(P^t)(A^t)
4) (A+B)^2
5) BAB

How would you approach this, are there any known relations between symmetric and anti symmetric matrices ? Thanks !

Just find the transpose of each of these matrices, using the facts that $A^{\small\mathsf{T}} = A$, $B^{\small\mathsf{T}} = -B$, and the transpose of a product is the product of the transposes in reverse order.

For example, the transpose of the matrix 1) is $$(5AB - 5BA)^{\small\mathsf{T}} = (5AB)^{\small\mathsf{T}} - (5BA)^{\small\mathsf{T}} = 5B^{\small\mathsf{T}}A^{\small\mathsf{T}} - 5A^{\small\mathsf{T}}B^{\small\mathsf{T}} = 5(-B)A - 5A(-B) = 5AB - 5BA.$$ That is the same as the original matrix, so you conclude that this matrix is symmetric.