Explain which have the same minimal polynomial

Reference no: EM13204368

1. Let F be a subfield of the complex numbers (or, a field of characteristic zero). Let V be a finite-dimensional vector space over F. Suppose that E1,...,Ek are projections of V and that E1+...+Ek = I. Prove that EiEj = 0 for i not equal to j (Hint: use the trace function and ask yourself what the trace of a projection is).
Hmm...the trace of a projection is its rank? But how does that help?

2. Give an example of two 4x4 nilpotent matrices which have the same minimal polynomial (they necessarily have the same characteristic polynomial) but which are not similar.

3. Let V be an n-dimensional vector space, and let T be a linear operator on V. Suppose that T is diagonalizable.

a) If T has a cyclic vector, show that T has n distinct characteristic values.

b) If T has n distinct characteristic values, and if {a1,...,an} is a basis of characteristic vectors for T, show that a = a1+...+an is a cyclic vector for T.

Reference no: EM13204368

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