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, $(AB^*)^* = BA^*\,$ .

Main Page : Linear Algebra

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