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Math Help - Linear Algebra Question

  1. #1
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    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?

    Many thanks for your consideration.
    CP
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  2. #2
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    Re: Linear Algebra Question

    Quote Originally Posted by CrispyPlanet View Post
    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~?

    You can read this page about skewed symmetric matrix.
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    Re: Linear Algebra Question

    Hi Plato

    Yes, that is what I meant, sorry.

    CP
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  4. #4
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    Re: Linear Algebra Question

    Quote Originally Posted by CrispyPlanet View Post
    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?

    Many thanks for your consideration.
    CP
    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*}.
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