This is a statement from wikipedia.I don't understand why can we deduce that $C$ is a partial isometry if $A^*A=B^*B$.
2026-03-26 08:31:19.1774513879
a question on partial isometry
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You have $$ \|CBx\|^2=\langle B^*C^*CBx,x\rangle=\langle A^*Ax,x\rangle=\langle B^*Bx,x\rangle=\|Bx\|^2. $$ So $C$ is isometric on $\operatorname{ran}B$, and so on $\overline{\operatorname{ran}B}$. On the orthogonal complement, it is $0$ by definition. So $C$ is a partial isometry.