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Consider the matrix A below. Find the eigenvalues of A using a calculator and use these to construct the characteristic polynomial of A, pA (x). State the algebraic multiplicity of each eigenvalue. Find all of the eigenspaces for A by computing expressions for null spaces, only using your calculator to row-reduce matrices. State the geometric multiplicity of each eigenvalue. Is A diagonalizable ? If not, explain why. If so, find a diagonal matrix D that is similar to A.
Physics
A calculator will report λ = 0 as an eigenvalue of algebraic multiplicity of 2, and λ =-1 as an eigenvalue of algebraic multiplicity 2 as well. Since eigenvalues are roots of the characteristic polynomial (Theorem EMRCP) we have the factored version
pA (x) = (x − 0)2(x − (−1))2 = x2(x2 + 2x + 1) = x4 + 2x3 + x2
The eigenspaces are then
Each eigenspace above is described by a spanning set obtained through
an application of Theorem BNS and so is a basis for the eigenspace. In each case the dimension,
and therefore the geometric multiplicity, is 2.
For each of the two eigenvalues, the algebraic and geometric multiplicities are equal. Theorem DMFE
says that in this situation the matrix is diagonalizable. We know from Theorem DC that when we
diagonal- ize A the diagonal matrix will have the eigenvalues of A on the diagonal (in some order).
So we can claim that
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