LA6

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Show that (AB^{*})^{*}=BA^{*}\,

Let A^{*}\, be the adjoint of the matrix A\,, and B^{*}\, be the adjoint of the matrix B\,. Then, for the matrix AB^{*}\,

\langle AB^{*}x,y\rangle =\langle B^{*}x,A^{*}y\rangle
=\langle x,(B^{*})^{*}A^{*}y\rangle
=\langle x,BA^{*}y\rangle

Therefore, BA^{*}\, is the adjoint of AB^{*}\,, or, in other words, (AB^{*})^{*}=BA^{*}\,.


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