How to find the partial isometry?

Question

Let $A$ and $B$ be two finite dimensional C* algebras, i.e. $A=\oplus^{m}_{k=1}M_{N_{k}}(\mathbb{C})$, $B=\oplus^{n}_{l=1}M_{K_{l}}(\mathbb{C})$. Let $\psi$, $\varphi:A\rightarrow B$ be two *-homomorphism, $\varphi_{l}$,$\psi_{l}$ be the $l$-component of $\varphi$ and $\psi$. Assume $Tr(\varphi_{l}(e_{11}^{(k)}))=Tr(\psi_{l}(e_{11}^{(k)}))$, where $e^{(k)}_{11}$ denotes the matrix $(e_{ij}^{(k)})_{pq}=\delta_{ip}\delta_{jq}$, $e_{11}^{(k)}\in M_{N_{k}}(\mathbb{C})$. I want to find partial isometry $w$ such that $w^{*}w=1-\varphi(1)$, $ww^{*}=1-\psi(1)$. I am totally clueless about the problem, I only know both $w^{*}w$ and $ww^{*}$ are projections. How should I approach?