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^*)\,


Main Page : Linear Algebra

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