# An inner product space over C

• May 19th 2010, 05:41 AM
Also sprach Zarathustra
An inner product space over C
Let V be an inner product space over field C(complex numbers). dim(V)<inf,
T:V-->V linear transformation.

1. Prove that T is normal (T*T=T*T) iff there exist polynomial p(x) in C[x] so that T*=p(T)

2. Conclude that if T,S:V-->V linear transformations, T normal and TS=ST, so T*S=S*T

Thank you all...
• May 19th 2010, 01:37 PM
Opalg
Quote:

Originally Posted by Also sprach Zarathustra
Let V be an inner product space over field C(complex numbers). dim(V)<inf,
T:V-->V linear transformation.

1. Prove that T is normal (T*T=T*T) iff there exist polynomial p(x) in C[x] so that T*=p(T)

If T is normal then it has a unitary diagonalisation. In other words, there is an orthonormal basis of V with respect to which T has a diagonal matrix (whose diagonal entries are the eigenvalues of T, but you don't need to know that for this question). Then the matrix of T* is also diagonal, and its diagonal entries are the complex conjugates of those of T. So you just need to show that there exists a polynomial taking a given finite set of complex numbers to their conjugates. If you don't know how to do that, do a Google search for "polynomial interpolation".

Quote:

Originally Posted by Also sprach Zarathustra
2. Conclude that if T,S:V-->V linear transformations, T normal and TS=ST, so T*S=S*T

This follows quite easily from 1. If S commutes with T, then it's easy to check that S commutes with any polynomial in T.
• May 19th 2010, 02:07 PM
Also sprach Zarathustra
Thank you very much!