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Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplicities for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so.
Physics
The characteristic polynomial of B is
= (−12 − x)(13 − x) − (30)(−5) Theorem DMST
= x2 − x − 6
= (x − 3)(x + 2)
From this we find eigenvalues λ = 3, −2 with algebraic multiplicities αB (3) = 1 and αB (−2) = 1.
For eigenvectors and geometric multiplicities, we study the null spaces of B − λI2 (Theorem EMNS).
Each eigenspace has dimension one, so we have geometric multiplicities γB (3) = 1 and γB (−2) = 1.
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