(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A n × n-matrix A satisfies the equation A^{2}= A.

(a) List all possible characteristic polynomials of A.

(b) Show that A is similar to a diagonal matrix

2. Relevant equations

3. The attempt at a solution

A^{2}= A

so, A^{2}- A = 0

A(A-I) = 0

Our minimal polynomial is x^{2}- x = m(x)

Our eigenvalues are 0 and 1, and since in our minimal polynomial each one has a multiplicity of 1, A is similar to a diagonal matrix consisting of n jordan blocks with either 1's or 0's on the diagonal and 0 everywhere else since we have an nxn matrix and n jordan blocks.

So I've proved part (b) first.

For part (a) we don't know how many times each eigenvalue occurs in the characteristic polynomial, so p(t) = t^{a}(t-1)^{b}

where a,b=0,1,2,...,n.

Is this correct?

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# Matrix, minimal polynomial

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