# HermitianSpaces: an Adjoint property proof (help)

1. Sep 6, 2008

### emlio

1. The problem statement, all variables and given/known data

$$V$$ is a linear space over $$C$$, finite n-dimensional

$$h: VxV \rightarrow C$$ is an Hermitian Product, POSITIVE DEFINED

and so

$$(V,h)$$ Hermitian Space
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$$L: V \rightarrow V$$, is a Linear Endomorphism of V

$$L^{*}$$ is the ADJOINT of $$L$$

$$h(L(v),w) = h(v,L^{*}(w)) \forall v,w\in V$$ (adjoint definition)

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Given:

$$B=\left\{v_{1},v_{2},v_{3},....................,v_{n}\right\}$$ is a h-ORTHORNORMAL basis for $$V$$

$$M\itshape^{B}_{B}(L)$$ i.e. matricial representation "from basis B to basis B" of L

$$M\itshape^{B}_{B}(L^{*})$$ i.e. matricial representation "from basis B to basis B" of L* (the adjoint of L)

I need to proof that:

$$M\itshape^{B}_{B}(L^{*})= (M\itshape^{B}_{B}(L))^{*}$$

2. Sep 6, 2008

### Dick

Start by writing down the expression for the ij_th element of both of those matrices (i.e. what is a general matrix element) in terms of the {v_i}. Now remember '*' of a matrix is the conjugate transpose.

3. Sep 6, 2008

### emlio

Tnx for the quick replay, I'll try as you say...

4. Sep 6, 2008

### emlio

I solved it !!!!!!!!!!!!!!!!:rofl:

$$B=\left\{v_{1},v_{2},v_{3},....................,v_{n}\right\}$$ is a h-ORTHORNORMAL basis for $$V$$

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$$M\itshape^{B}_{B}(L)=(a_{x,y})$$ j-th Columns is:

$$L(v_{j})= a_{1j}v_{1} +....+a_{nj}v_{n}$$

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$$M\itshape^{B}_{B}(L^{*})=(b_{x,y})$$ k-th COLUMN IS:

$$L^{*}(v_{k})= b_{1j}v_{1}+....+b_{nk}v_{n}$$

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$$h(L(v_{j}),v_{k}) = h(v_{j},L^{*}(v_{k}))$$

$$h( a_{1j}v_{1} +....+a_{nj}v_{n} , v_{k}) = h(v_{j}, b_{1j}v_{1}+....+b_{nk}v_{n})$$

................

applying properties of "h"scalar product and properties of h-ohrtonormal bases

...............
$$a_{jk}*h(v_{k},v_{k}) = \bar{b}_{kj}*h(v_{j},v_{j})$$

$$a_{jk} = \bar{b}_{kj}$$

$${b}_{jk} = \bar{a}_{kj}$$ !

5. Sep 6, 2008

### emlio

There are some typing errors, I've written "j" instead of "k" in some parts, the proof is ok

6. Sep 6, 2008

### Dick

Looks like the idea is right. Good job.