Vectors and matrices. Let C denote the field of complex numbers and let C ⁿ be the vector space of column vectors

We define an inner product and a norm by

Here u _j is the complex conjugate of u _j . If A=( a _jk) ? C ^n,n is a complex n by n matrix, then

denotes the complex conjugate transpose of A.

It holds that

The spectral-norm of A is

One can show that

where ?(B) denotes the spectral radius of a matrix B, i.e., the largest absolute value of all eigenvalues of B. A matrix U ? C ^{n , n} is called unitary if

If U is unitary then, for any u ? C ⁿ,

and therefore

A matrix U with columns u ¹ , , u ⁿ ? C ⁿ is unitary if and only if

i.e., the columns form an orthonormal system.

An important result from linear algebra is

Schur s Theorem. Let A ? C ^n,n denote a matrix with eigenvalues ? ₁, , ? _n in any prescribed order. There is a unitary matrix U such that

is upper triangular with diagonal entries r _jj = ? _j, j=1, , n.

Proof. We use induction on n; the case n=1 is trivial. Let Au ¹ = ? ₁