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    Diagonalizability of a $4\times4$ matrix with minimal polynomial $x^2(x-1)^2$

    E-HK-63-4

    Area
    Algebra
    Topics
    Annihilating Polynomials, Linear Algebra
    Status
    draft

    Is the matrix \(A\) of Exercise 3 similar over the field of complex numbers to a diagonal matrix?

    Solution

    No. A diagonalizable matrix has a minimal polynomial which is a product of distinct linear factors: if \(A=SDS^{-1}\) with \(D\) diagonal, then the product of \(x-\lambda\) over the distinct diagonal entries annihilates \(D\), hence \(A\).

    By Exercise 3, \[ m_A(x)=x^2(x-1)^2, \] which has repeated factors. Therefore \(A\) is not diagonalizable over \(\mathbb C\).

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