# Matrix/basis question

• Dec 22nd 2010, 12:09 PM
worc3247
Matrix/basis question
I can see that this is true but I am having trouble presenting a convincing argument.
Let V = $\displaystyle M_{n\times n} (\mathbb{R})$. Show that V has a basis with the property that each matrix in the basis is either symmetric or skew symmetric.

The second part of the question however I have no idea where to start:
Show also that V has a basis with the property that each matrix in the basis is an invertible matrix.
• Dec 22nd 2010, 02:55 PM
FernandoRevilla
Quote:

Originally Posted by worc3247
I can see that this is true but I am having trouble presenting a convincing argument.
Let V = $\displaystyle M_{n\times n} (\mathbb{R})$. Show that V has a basis with the property that each matrix in the basis is either symmetric or skew symmetric.

If:

$\displaystyle F_1=\left\{{A\in{V}:A^t=A}\right\},\quad F_2=\left\{{A\in{V}:A^t=-A}\right\}$

then,

$\displaystyle V=F_1\oplus F_2$

so, the union of a basis of $\displaystyle F_1$ with a basis of $\displaystyle F_2$ is a basis of $\displaystyle V$.

Fernando Revilla
• Dec 23rd 2010, 12:46 AM
Opalg
Quote:

Originally Posted by worc3247
Let V = $\displaystyle M_{n\times n} (\mathbb{R})$.

...

I have no idea where to start:
Show also that V has a basis with the property that each matrix in the basis is an invertible matrix.

Start with a basis for V in which the first element in the basis is the identity matrix $\displaystyle I$. You can add a multiple of $\displaystyle I$ to each other element in the basis so as to make that element invertible, and the resulting set of matrices will still be a basis.