Positive matrices

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Section 10.1 Positive matrices

We’re now going to look at the notion of positive operators inside \(\Lop\text{.}\) In \(\R\text{,}\) non-negative elements \(a\geq 0\) have the property that there exist non-negative real numbers \(b\) such that \(b^2 = a\text{.}\) We usually call \(b = \sqrt{a}\text{.}\)

In \(M_n(\C)\text{,}\) suppose that \(A\) is a matrix so that \(\ip{Ax}{x} \geq 0\) for all \(x \in \C^n\text{.}\) Such matrices are called positive, and we denote this by \(A \geq 0\text{.}\) Then

\begin{equation*} \ip{Ax}{x} = \cc{\ip{x}{Ax}} = \ip{A\ad x}{x} \end{equation*}

so for all \(x \in \C^n\text{,}\) we have that \(\ip{(A - A\ad)x}{x} \geq 0\text{,}\) which implies that \(A = A\ad\text{.}\) Because self-adjoint matrices have unitary dialgonalizations and real eigenvalues, the positivity condition implies that all of the eigenvalues of \(A\) are non-negative (as \(\ip{Av}{v} = \ip{\la v}{v} = \la \geq 0\text{.}\)) So if \(A \geq 0\text{,}\) then there exist a unitary matrix \(U\) and a diagonal matrix \(D\) with real non-negative entries \(\la_i\) on the diagonal so that \(A = U\ad D U\text{.}\) Now construct the matrix

\begin{equation*} B = U\ad \bpm \sqrt{\la_1} \amp \amp \amp \\ \amp \sqrt{\la_2} \amp \amp \\ \amp \amp \ddots \amp \\ \amp \amp \amp \sqrt{\la_n} \epm U. \end{equation*}

Then \(B^2 = A\text{,}\) and \(B \geq 0\text{.}\) We call the matrix \(B\) the square root of \(A\) and denote it \(B = \sqrt{A}\text{.}\)

We can use the same sort of ideas to define and identify positive operators, which turn out to play a key role in operator theory. We will (hopefully) try to use these ideas to solve a classical problem in function theory, the Nevanlinna-Pick interpolation problem.

Question 10.1.1. Nevanlinna-Pick problem.

Let \(\D\) denote the complex unit disk. Given points \(\la_1, \ldots, \la_n\) and \(z_1, \ldots, z_n\) in \(\D\text{,}\) when does there exist an analytic function \(f\) so that \(f(\la_i) = z_i\) for all \(i\) and \(\norm{f} \leq 1.\) where the norm is the sup-norm on the disk.

Analytic functions of norm 1 on the disk are called the Schur class.