Math Help - Linear Algebra Question

1. Linear Algebra Question

Question: Let A be an nxn matrix and let:

B=A + B^T and C=A - A^T

a) Show that B is symmetric and C is skewed symmetric.
b) Show that every nxn matrix can be represented as a sum of a symmetric matrix and a skew symmetric matrix.

I'm not certain how to tackle this. Firstly, the book I am using does not define the term 'skewed symmetry'. What does it mean? Any helps/tips/prompts etc most welcome.

On a more general note, I've just started linear algebra and can see myself getting quite overwhelmed as I lack the intuitive ability to work well with matrices. Can any of you fine gentlemen give me any tips for keeping afloat?

CP

2. Re: Linear Algebra Question

Originally Posted by CrispyPlanet
Question: Let A be an nxn matrix and let:

B=A + B^T and C=A - A^T
Do you mean $B=A+A^T~?$

3. Re: Linear Algebra Question

Hi Plato

Yes, that is what I meant, sorry.

CP

4. Re: Linear Algebra Question

Originally Posted by CrispyPlanet
Question: Let A be an nxn matrix and let:

B=A + B^T and C=A - A^T

a) Show that B is symmetric and C is skewed symmetric.
b) Show that every nxn matrix can be represented as a sum of a symmetric matrix and a skew symmetric matrix.

I'm not certain how to tackle this. Firstly, the book I am using does not define the term 'skewed symmetry'. What does it mean? Any helps/tips/prompts etc most welcome.

On a more general note, I've just started linear algebra and can see myself getting quite overwhelmed as I lack the intuitive ability to work well with matrices. Can any of you fine gentlemen give me any tips for keeping afloat?

A matrix is symmetric if it is equal to its transpose, i.e. \displaystyle \begin{align*} A = A^T \end{align*}.
A matrix is skew-symmetric if its transpose is also its negative, i.e. \displaystyle \begin{align*} A^T = -A \end{align*}.
It might also help you to know that \displaystyle \begin{align*} \left( A + B \right) ^T = A^T + B^T \end{align*} and \displaystyle \begin{align*} \left( MN \right) ^T = N^TM^T \end{align*}.