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Given two matrices:

$$ A= \begin{pmatrix} 1 & 0 & 1 & 2\\ 2 & 1 & -1 & 1\\ 0 & 1 & 0 & 2\\ \end{pmatrix} $$

$$ B= \begin{pmatrix} 2 & -1 & -2 \\ 3 & 4 & -3 \\ 0 & 1 & 2 \\ \end{pmatrix} $$

a) Construct two symmetric square matrices from matrix $A$
b) Calculate the symmetric and anti-symmetric part of matrix $B$

I don't know how to go about question a). I tried a LU Decomposition but it didn't give the desired outcome. Only other thing I could think of was to take the Lower matrix from the LU decomposition, then add it to its transpose, which of course gives a symmetric square matrix but it seemed too "forced" to me, plus it only gives one matrix instead of the two that are demanded. So I'm sure that there's another method that I'm not aware of.

As for question b) I don't understand what they mean by "symmetric and anti-symmetric parts" and what does it mean to calculate them.

Hope someone can help me solve this, or at least give me some hints. Thanks.

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    $\begingroup$ question a) is really ambiguous, do you have more context about it? $\endgroup$ May 8 '18 at 14:52
  • $\begingroup$ No I don't. That's all that was given in the question. What's one way you would go about it if you don't mind me asking? $\endgroup$
    – Metrician
    May 8 '18 at 15:10
  • $\begingroup$ in real life? ask the professor $\endgroup$ May 8 '18 at 15:28
  • $\begingroup$ @GuillermoMosse That would've been my go-to but it wasn't given by a professor I just stumbled upon it while preparing for an upcoming entry exam $\endgroup$
    – Metrician
    May 8 '18 at 15:56
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    $\begingroup$ My guess for part (a) is that you’re simply meant to compute $A^TA$ and $AA^T$. $\endgroup$
    – amd
    May 8 '18 at 17:59
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For b): Prove that $B= \frac{1}{2}(B+B^t)+\frac{1}{2}(B-B^t)$

This decomposition can be applied to all square matrices. What are the properties of the matrices that are summated?

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    $\begingroup$ It's apparent that I don't yet have a deep understanding about this particular decomposition. Can you please tell me what it's called so I can study it thoroughly? $\endgroup$
    – Metrician
    May 8 '18 at 15:10
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    $\begingroup$ analyze the decomposition and see how it relates to your problem $\endgroup$ May 8 '18 at 15:30
  • $\begingroup$ Alright. Thanks for the answer. $\endgroup$
    – Metrician
    May 8 '18 at 15:56
  • $\begingroup$ Then accept it as an answer! You have to click on the "tick" at the left of the answer. (Do you want more details? Did you reach a solution?) $\endgroup$ May 15 '18 at 18:10

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