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Math Help - Diagonalizable matrix

  1. #1
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    Diagonalizable matrix

    A scalar matrix is a square matrix of the form \lambda\I for some scalar \lambda\ that is, a scalar matrix is a diagonal matrix in which all the entries are equal.

    a) Prove that if a square matrix A is similar to a scalar matrix \lambda\I, Then A= \lambda\I

    b)
    Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix.

    c) Prove that \begin{pmatrix}\ 1 & 1 \\ 0 & 1 \end{pmatrix}\ is not diagonalizable.
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  2. #2
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    Hello,

    a) If there is an invertible matrix P such that P^{-1}AP=\lambda I, then A=P \lambda IP^{-1}=\lambda I.

    b) If A is diagonalizable, there exists an invertible matrix P such that
    P^{-1}AP=\text{diag}(\lambda_1, \lambda_2,\ldots, \lambda_n)
    where \lambda_1,\ldots,\lambda_n are the eigenvalues.
    If A has only one eigenvalue,
    \lambda_1=\cdots =\lambda_n.
    By a), A is a scalar matrix.

    c) It is not a scalar matrix, but 1 is its only eigenvalue. Use b).

    Bye
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