LA9

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Show that the zero matrix {\mathcal  {O}}\, is self-adjoint.

\langle {\mathcal  {O}}x,y\rangle =\langle {\vec  {0}},y\rangle
=0\,

and

\langle x,{\mathcal  {O}}y\rangle =\langle x,{\vec  {0}}\rangle
=0\,

Thus, \langle {\mathcal  {O}}x,y\rangle =\langle x,{\mathcal  {O}}y\rangle . Therefore, {\mathcal  {O}}\, is the adjoint of {\mathcal  {O}}\,, or, in other words, {\mathcal  {O}}\, is self-adjoint.


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