Paper 1, Section II, 6A\mathbf{6 A}

Vectors and Matrices | Part IA, 2020

What does it mean to say an n×nn \times n matrix is Hermitian?

What does it mean to say an n×nn \times n matrix is unitary?

Show that the eigenvalues of a Hermitian matrix are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal.

Suppose that AA is an n×nn \times n Hermitian matrix with nn distinct eigenvalues λ1,,λn\lambda_{1}, \ldots, \lambda_{n} and corresponding normalised eigenvectors u1,,un\mathbf{u}_{1}, \ldots, \mathbf{u}_{n}. Let UU denote the matrix whose columns are u1,,un\mathbf{u}_{1}, \ldots, \mathbf{u}_{n}. Show directly that UU is unitary and UDU=AU D U^{\dagger}=A, where DD is a diagonal matrix you should specify.

If UU is unitary and DD diagonal, must it be the case that UDUU D U^{\dagger} is Hermitian? Give a proof or counterexample.

Find a unitary matrix UU and a diagonal matrix DD such that

UDU=(203i0203i02)U D U^{\dagger}=\left(\begin{array}{ccc} 2 & 0 & 3 i \\ 0 & 2 & 0 \\ -3 i & 0 & 2 \end{array}\right)

Typos? Please submit corrections to this page on GitHub.