LA10

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Show that (\alpha A+\beta B)^{*}=(\overline {\alpha }A^{*}+\overline {\beta }B^{*})\,

Let A^{*}\, be the adjoint of the matrix A\,, and B^{*}\, be the adjoint of the matrix B\,. Then, for the matrix (\alpha A+\beta B)\,

\langle (\alpha A+\beta B)x,y\rangle \, =\langle \alpha Ax+\beta Bx,y\rangle \,
=\langle \alpha Ax,y\rangle +\langle \beta Bx,y\rangle \,
=\alpha \langle Ax,y\rangle +\beta \langle Bx,y\rangle \,
=\alpha \langle x,A^{*}y\rangle +\beta \langle x,B^{*}y\rangle \,
=\langle x,\overline {\alpha }A^{*}y\rangle +\langle x,\overline {\beta }B^{*}y\rangle \,
=\langle x,\overline {\alpha }A^{*}y+\overline {\beta }B^{*}y\rangle \,
=\langle x,(\overline {\alpha }A^{*}+\overline {\beta }B^{*})y\rangle \,

Therefore, (\overline {\alpha }A^{*}+\overline {\beta }B^{*})\, is the adjoint of (\alpha A+\beta B)\,, or, in other words, (\alpha A+\beta B)^{*}=(\overline {\alpha }A^{*}+\overline {\beta }B^{*})\,


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