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Let $A$ be a real matrix, not necessarily square.

a) Show that both $A^{*} A$ and $A A^{*}$ are self-adjoint.
b) Show that both $A^{*} A$ and $A A^{*}$ are positive semi-definite.
c) Show that $\operatorname{ker} A=\operatorname{ker} A^{*} A$. [HINT: Show separately that $\operatorname{ker} A \subset \operatorname{ker} A^{*} A$ and ker $A \supset \operatorname{ker} A^{*} A$. The identity $\left\langle\vec{x}, A^{*} A \vec{x}\right\rangle=\langle A \vec{x}, A \vec{x}\rangle$ is useful.]
d) If $A$ is one-to-one, show that $A^{*} A$ is invertible
e) If $A$ is onto, show that $A A^{*}$ is invertible.
f) Show that the non-zero eigenvalues of $A^{*} A$ and $A A^{*}$ agree. Generalize.
g) Show that image $\left(A A^{*}\right)=\left(\operatorname{ker} A A^{*}\right)^{\perp}=\left(\operatorname{ker} A^{*}\right)^{\perp}=\operatorname{image} A$.
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