A First Course in Linear Algebra · Problem C20

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · diagonalization

Problem

Consider the matrix AA below. First, show that AA is diagonalizable by computing the geometric multiplicities of the eigenvalues and quoting the relevant theorem. Second, find a diagonal matrix DD and a nonsingular matrix SS so that S−1AS=DS^{-1}AS = D. (See Exercise EE.C20 for some of the necessary computations.) A=[18−1533−15−48−66−99−1695−69−4]A = \begin{bmatrix} 18 & -15 & 33 & -15 \\ -4 & 8 & -6 & 6 \\ -9 & 9 & -16 & 9 \\ 5 & -6 & 9 & -4 \end{bmatrix}

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