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An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose.

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Solution for $Xa + X^Tb = c$ where $X^TX = I$?

As a beginning of a search for solutions, we can take norms of each side. We then get $$\|a\|^2 + \|b\|^2 + 2 b\cdot (X^2 a)=\|c\|^2,$$ which is enough to fix the angle between $b$ and $X^2 a$. Simil …
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