Proof regarding skew symmetric matrices

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Homework Help Overview

The problem involves proving that if a matrix A is skew symmetric, then A raised to any positive odd integer k is also skew symmetric. The discussion centers around properties of skew symmetric matrices and potential proof techniques.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to prove the statement, suggesting that induction might be involved. Some participants clarify the definition of skew symmetric matrices and propose using properties of transposes to show that specific powers, like A^3, are skew symmetric. Others question whether induction is necessary or if simpler reasoning could suffice.

Discussion Status

Participants are actively exploring different methods to approach the proof, including induction and properties of matrix transposes. There is a recognition of the potential complexity of the problem, but no consensus has been reached on a definitive method.

Contextual Notes

There is a focus on proving the statement specifically for positive odd integers, and participants are considering the implications of this constraint in their discussions.

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Homework Statement



Show that if A is skew symmetric, then Ak is skew symmetric for any positive odd integer k.

Homework Equations





The Attempt at a Solution



Wow, I have no idea how to prove this. I'm guessing there's going to be induction involved. I know that the base case of k = 1 is true, because A1 = A, where A is skew symmetric. But after that, I don't know what to do at all.

Any suggestions on where to go from here?
 
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Skew-symmetric means A^(T)=(-A). (T=transpose). Does that help? Can you use that to show A^3 is skew-symmetric?
 
Dick said:
Skew-symmetric means A^(T)=(-A). (T=transpose). Does that help? Can you use that to show A^3 is skew-symmetric?

[tex] \begin{align*}<br /> A &= -A^T\\<br /> A^3 &= (-A^T)^3\\<br /> &= (-A^T)(-A^T)(-A^T)\\<br /> &= (A^2)^T(-A^T)\\<br /> &= (-AA^2)^T\\<br /> &= (-A^3)^T<br /> \end{align*}[/tex]

so A3 is skew symmetric.

Thanks again for your help Dick. Now, how can I go about proving this for the rest of the positive odd integers. Will induction work? I'm trying to think of a way to do it, but I'm stuck on the "odd" integers part.
 
Sure, induction will work. But that's probably overkill. (A^n)^T=(A^T)^n. Just factor out the (-1)^n.
 
Dick said:
Sure, induction will work. But that's probably overkill. (A^n)^T=(A^T)^n. Just factor out the (-1)^n.

OH...haha thanks very much. I tend to over analyze sometimes...
 

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