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Definition of notation:

First let x for this problem denote the elementart tensor product.

Then let (AxB) acting on a vector v be defined by A<B,V> where this is the standard inner product.

Problem statement

With those defintions consider e and v members of Rn the vector space of N tuples allow A to be defined as A = exf +fxe

find the

eigen vectors

eigen spaces

specteral decompostition

Relevent equations

let the matrix that defines AxB be defined by (aibj) where ai and bj are the componets of the vectors

Attempt at a soloution

solve for Ag=g*lambda where we know g must be in the span{e,f}

we know that there are only 2 non-zero eigen values because there are N-2 mutually orthogonal components to e and f.

we can also know that all eigan values are real.

Thanks for any help