Week of 1/21/08

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Revision as of 15:37, 21 January 2008

It's a matter of convention that the anti-linear term is the second one in the inner product:

$\langle u+v,w \rangle \equiv \langle u,w \rangle + \langle v,w \rangle$

$\langle u,v+w \rangle \equiv \langle u,v \rangle + \langle u,w \rangle$

$\langle \alpha u,v \rangle \equiv \alpha \langle u,v \rangle$

$\langle u,\alpha v \rangle \equiv \alpha^* \langle u,v \rangle$

$\langle u,v \rangle \equiv \langle v,u \rangle ^*$

$\langle u,u \rangle = 0,$ with equality only if $u \equiv 0$

The basic example is the form

$h(z,w) \equiv \sum z_i w^* _i$

NB if $z = x + I y$ , then $z * z = (x - I y)(x + I y) = x 2 + y 2 = z z *$

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 [http://mathworld.wolfram.com/HermitianInnerProduct.html Definition of complex (Hermitian) inner product]
+It's a matter of convention that the anti-linear term is the second one in the inner product:
 <math> \langle u+v,w \rangle \equiv \langle u,w \rangle + \langle v,w \rangle </math>
@@ Line 16: / Line 19: @@
 <math>h(z,w) \equiv \sum z_i w^* _i</math>
+NB if <math>z = x + I y</math>, then <math>z^* z = (x - I y)(x + I y) = x^2 + y^2 = z z^*</math>