An inner product space over C

Dec 2009
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434
Russia
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...
 

Opalg

MHF Hall of Honor
Aug 2007
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Leeds, UK
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".

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.