Property of positive matrix

Question

Let $A$ be a $C^*$-algebra and let $a = \begin{pmatrix}a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2}& \dots & a_{nn}\end{pmatrix} \in M_n(A)$ be a positive matrix.

Is it true that if $\lambda_1, \dots, \lambda_n \in \mathbb{C}$, then $$\begin{pmatrix}\overline{\lambda}_1 & \overline{\lambda}_2 & \dots & \overline{\lambda}_n\end{pmatrix}\begin{pmatrix}a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2}& \dots & a_{nn}\end{pmatrix}\begin{pmatrix}\lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n\end{pmatrix}$$ is a positive element of $A$?

I tried using the characterisation $a= x^*x$ but the computation became quite ugly so I was wondering if there is a conceptual easy way to see this.

Answer 1

If $a$ is a positive element of $M_n(A)$, then there is a $b \in M_n(A)$ with $a = b^*b$. It follows that $$ \begin{pmatrix}\overline{\lambda}_1 & \overline{\lambda}_2 & \dots & \overline{\lambda}_n\end{pmatrix}\begin{pmatrix}a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2}& \dots & a_{nn}\end{pmatrix}\begin{pmatrix}\lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n\end{pmatrix} = \left[b\begin{pmatrix}\lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n\end{pmatrix} \right]^*\left[b\begin{pmatrix}\lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n\end{pmatrix}\right]. $$ Now, write $$ b \pmatrix{\lambda_1\\ \vdots \\ \lambda_n} = \pmatrix{c_1\\ \vdots \\ c_n} $$ for some $c_1,\dots,c_n \in A$. From the above, we have $$ \begin{pmatrix}\overline{\lambda}_1 & \overline{\lambda}_2 & \dots & \overline{\lambda}_n\end{pmatrix}\begin{pmatrix}a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2}& \dots & a_{nn}\end{pmatrix}\begin{pmatrix}\lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n\end{pmatrix} = \pmatrix{c_1 \\ \vdots \\ c_n}^* \pmatrix{c_1 \\ \vdots \\ c_n} = \sum_{i=1}^n c_i^*c_i. $$ Now, it suffices to note that the sum of positive elements is positive.