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Tag Archives: Linear Algebra
Consider the real numbers for . Prove that
Using the identity in combination with we have:
Let such that
Prove that if is odd.
Let . Then
Taking conjugate transpose we also have that
Hence . However it also holds . Combiming these two we get that
If we are done. Otherwise is real. In that case we have
since is odd. Hence as wanted.
Let be unitary linearly independent vectors. Evaluate the determinant
where denotes the outer product.
Let and such that . Prove that
Since is real, its complex eigenvalues come in conjugate pairs. Thus, in this case we conclude that has eigenvalues .
Now, if is an eigenvalue of , then is an eigenvalue of . Thus, the matrix has eigenvalues and .
Now, is the product of these eigenvalues, which is to say
Let . Prove that .
The characteristic polynomial of is . This in return means and . Thus,