Question

For each of the following matrices A ? Mnxn (F), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue ? of A, find the set of eigenvectors corresponding to ?. (iii) If possible, find a basis for F<sup>n</sup> consisting of eigenvectors of A. (iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q<sup>-1</sup>AQ = D. 258 (a) A = \begin{pmatrix} 1 & 2\\ 3 & 2 \end{pmatrix} for F = R (b) A = \begin{pmatrix} 0 & -2 & -3\\ -1 & 1 & -1\\ 2 & 2 & 5 \end{pmatrix} for F = R (c) A = \begin{pmatrix} i & 1\\ 2 & -i \end{pmatrix} for F = C (d) A = \begin{pmatrix} 2 & 0 & -1\\ 4 & 1 & -4\\ 2 & 0 & -1 \end{pmatrix} for F = R

          For each of the following matrices A ? Mnxn (F),
(i) Determine all the eigenvalues of A.
(ii) For each eigenvalue ? of A, find the set of eigenvectors corresponding to ?.
(iii) If possible, find a basis for F<sup>n</sup> consisting of eigenvectors of A.
(iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q<sup>-1</sup>AQ = D.
258
(a) A = \begin{pmatrix} 1 & 2\\ 3 & 2 \end{pmatrix} for F = R
(b) A = \begin{pmatrix} 0 & -2 & -3\\ -1 & 1 & -1\\ 2 & 2 & 5 \end{pmatrix} for F = R
(c) A = \begin{pmatrix} i & 1\\ 2 & -i \end{pmatrix} for F = C
(d) A = \begin{pmatrix} 2 & 0 & -1\\ 4 & 1 & -4\\ 2 & 0 & -1 \end{pmatrix} for F = R

For each of the following matrices A ? Mnxn (F),
(i) Determine all the eigenvalues of A.
(ii) For each eigenvalue ? of A, find the set of eigenvectors corresponding to ?.
(iii) If possible, find a basis for F<sup>n</sup> consisting of eigenvectors of A.
(iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q<sup>-1</sup>AQ = D.
258
(a) A =
< p m a t r i x >
for F = R
(b) A =
< p m a t r i x >
for F = R
(c) A =
< p m a t r i x >
for F = C
(d) A =
< p m a t r i x >
for F = R

Added by Victor B.

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