Question

2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7).

Answer 1

We know that the characteristic equation can be written as

$\det(A-xI) = x^n - tr(A)x^{n-1}+\dots+(-1)^n \det(A)$

We also know that the eigenvalues of A are solution of this equation. Hence, from this, we get

tr(A) = 11 + 12 + ... + In det(A) = \112 ... In

Now,we only have to show that
tr(A* A) = 1112 + 4212 + ... + An
. To show this, we will show that the eigenvalues of $(A^*A)$ are of the form $|\lambda_i|^2$ . Now, to show that, let us assume that x is an eigenvector. Then, we get

$A^*Ax = \lambda x \\ \Rightarrow x^*A^*Ax = \lambda x^*x \\ \Rightarrow (Ax)^*(Ax) = \lambda x^*x \\ \Rightarrow |Ax|^2 = \lambda |x|^2 \\ \Rightarrow |Ax| = \sqrt\lambda |x| \\ \Rightarrow Ax = \sqrt\lambda x \Rightarrow \sqrt\lambda = |\lambda_i| \Rightarrow \lambda = |\lambda_i|^2$

Hence, from this, we are done