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The questions this week is about diagonalizability of matrices when we have fewer than n different...

The questions this week is about diagonalizability of matrices when we have fewer than n different eigenvalues. Recall the following facts as starting points:

• An n × n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.

• Eigenvectors with different eigenvalues must be linearly independent.

• The number of times an eigenvalue appears as a root of the characteristic polynomial is at least the dimension of the corresponding eigenspace, and the total degree of the characteristic polynomial is n.

(2) Suppose we know a 4 × 4 matrix has eigenvalues λ = 1, 3, −2, and each eigenspace has dimension 2. Why do we know we have lost our minds and gone crazy? (In other words, why is this impossible?)

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