Inner product space

**vincisonfire** · March 23rd 2009, 03:51 PM

Show that if $V = \mathbb C^n$
, then for any inner product on V , there
is a unique complex n × n matrix A such that $\langle \vec{v},\vec{w}\rangle =\vec{v}^T A \overline{\vec{w}}$ for
all $\ \vec{v},\vec{w}\in V$ .
How to show existence?

**siclar** · March 23rd 2009, 04:40 PM

What happens to the usual basis vectors, i.e. $<e_i,e_j>$ where $e_i$ is the vector with 1 in the ith coordinate and 0 elsewhere? Note that, assuming such a representation is always possible, $<e_i,e_j>=a_{ij}$ .

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