Math Problems Vault

Problem VII.4.1

(a)(b)

In a two-dimensional unitary space an orthonormal basis is given, together with vectors $a$ and $b$ whose coordinates in this basis are respectively $1+i, 1-i$ and $-i, 2-2 i$ .

?

(a)

Find their lengths, and also the cosines of the angles between $a$ and $b$ and between $b$ and $a$ .

(b)

Orthogonalize this pair of vectors.

Problem VII.4.2

Write down some non-real Hermitian matrix and some non-real unitary matrix of order 3.

?

Problem VII.4.3

Prove that the roots of the characteristic equation of a real orthogonal matrix (including the complex ones) are equal to 1 in modulus.

?

Problem VII.4.4

Find an orthonormal basis of eigenvectors and the matrix of the transformation in this basis for the transformation $\mathrm{A}$ given in an orthonormal basis by the matrix

\left\| \begin{array}{cc} 0 & i \\ -i & 0 \end{array}\right\| .

Is the transformation self-adjoint, unitary?

?

Problem VII.4.5

Find an orthonormal basis of eigenvectors of the unitary transformation given in an orthonormal basis by the matrix

A=\left\| \begin{array}{cr} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right\| .

?