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MathsGee Android Q&A

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Let \(A\) and \(B\) be \(n \times n\) complex matrices that commute: \(A B=B A\). If \(\lambda\) is an eigenvalue of \(A\), let \(\mathcal{V}_{\lambda}\) be the subspace of all eigenvectors having this eigenvalue.
a) Show there is an vector \(v \in \mathcal{V}_{\lambda}\) that is also an eigenvector of \(B\), possibly with a different eigenvalue.
b) Give an example showing that some vectors in \(\mathcal{V}_{\lambda}\) may not be an eigenvectors of B.
c) If all the eigenvalues of \(A\) are distinct (so each has algebraic multiplicity one), show that there is a basis in which both \(A\) and \(B\) are diagonal. Also, give an example showing this may be false if some eigenvalue of \(A\) has multiplicity greater than one.
in Mathematics by Platinum (147,718 points) | 282 views

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MathsGee Android Q&A

MathsGee Android Q&A