# positive matrix

• Oct 16th 2010, 01:10 AM
Mauritzvdworm
positive matrix
Say we have an n-dimensional row vector $a=(a_1,\dots,a_n)$ with entries in a C*-algebra $A$. We construct the matrix
$a^*a=
\left(
\begin{array}{lll}
a_1^*a_1 &\dots &a_1^*a_n \\
\vdots & \ddots & \vdots\\
a_n^*a_1 & \dots & a_n^*a_n
\end{array}
\right)
$

How do we show that this matrix is positive?

Here is my idea:
If we have a representation $\varphi:A\rightarrow B(H)$ for some Hilbert space $H$, we can inflate this homomorphism to
$\varphi:M_n(A)\rightarrow M_n(B(H))$ with norm $\|T\|=\|\varphi(T)\|$ which makes $M_n(A)$ into a C*-algebra.
Furthermore we know that $M_n(B(H))$ is isomorphic to $B(H^{(n)})$ where $H^{(n)}:=\oplus^n_{i=1}H$.

We know that $\varphi(a^*a)^*=\varphi(a^*a)$, so it is self adjoint. Now is we can show that $\langle \varphi(a^*a)x,x\rangle\geq 0$ where $x\in H^{(n)}$ then the matrix is positive.

So
$$\langle \varphi(a^*a)x,x\rangle= \langle \left( \begin{array}{lll} \varphi(a_1)^*\varphi(a_1) &\dots &\varphi(a_1^*)\varphi(a_n) \\ \vdots & \ddots & \vdots\\ \varphi(a_n)^*\varphi(a_1) & \dots & \varphi(a^*_n)\varphi(a_n) \end{array} \right)x,x\rangle\\ =\sum^{n}_{i=1}\sum^{n}_{j=1}\langle \varphi(a_i)^*\varphi(a_j)x_j,x_i\rangle$$

so when $i=j$ we have positivity, but what about the terms where $i\neq j$?

Or am I completely on the wrong track?