91Ó°ÊÓ

For the complex numbers \(3+4 i\) and \(1-i\), (a)' find their positions in the complex plane. (b) find their sum and product. (c) find their conjugates and their absolute values. Do the original numbers lie inside or outside the unit circle?

(a) How many degrees of freedom are there in a real symmetric matrix, a real diagonal matrix, and a real orthogonal matrix? (The first answer is the sum of the other two, because \(A=Q \Lambda Q^{\mathrm{T}}\).) (b) Show that 3 by 3 Hermitian matrices \(A\) and also unitary \(U\) have 9 real degrees of freedom (columns of \(U\) can be multiplied by any \(e^{i \theta}\) ).

Prove that every unitary matrix \(A\) is diagonalizable, in two steps: (i) If \(A\) is unitary, and \(U\) is too, then so is \(T=U^{-1} A U\). (ii) An upper triangular \(T\) that is unitary must be diagonal. Thus \(T=\Lambda\). Any unitary matrix \(A\) (distinct eigenvalues or not) has a complete set of orthonormal eigenvectors. All eigenvalues satisfy \(|\lambda|=1\).

Which classes of matrices does \(P\) belong to: orthogonal, invertible, Hermitian, unitary, factorizable into \(L U\), factorizable into \(Q R\) ? $$ P=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right] $$