Proving Unique Decomposition of a Square Matrix

mlarson9000
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Homework Statement



An nxn matrix C is skew symmetric if C^t = -C. Prove that every square matrix A can be written uniquely as A = B + C where B is symmetric and C is skew symmetric.


Homework Equations





The Attempt at a Solution



No clue.
 
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You aren't trying very hard, now are you? Tell me about (A+A^T) and (A-A^T). Are they symmetric, skew-symmetric or neither? You have to help here.
 
(A+A^T)is symmetric, (A-A^T)is skew symmetric, but adding them together produces 2A, not A. I'm not sure what to do with this information.
 
mlarson9000 said:
(A+A^T)is symmetric, (A-A^T)is skew symmetric, but adding them together produces 2A, not A. I'm not sure what to do with this information.

That's not such a big problem. Divide each one by two.
 
Dick said:
That's not such a big problem. Divide each one by two.

How embarrassing.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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