91Ó°ÊÓ

Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and (d) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{rrr} 1 & -1 & -1 \\ 0 & 2 & 0 \\ -1 & -1 & 1 \end{array}\right]$$

Use the power method to approximate the dominant eigenvalue and eigenvector of \(A\). Use the given initial vector \(\mathbf{x}_{0},\) the specified number of iterations \(k,\) and three-decimal-place accuracy. $$A=\left[\begin{array}{rr} 14 & 12 \\ 5 & 3 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], k=5$$

Show that \(\lambda\) is an eigenvalue of \(A\) and find one eigenvector corresponding to this eigenvalue. $$A=\left[\begin{array}{rr} 0 & 4 \\ -1 & 5 \end{array}\right], \lambda=1$$

Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and (d) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{rrrr} 3 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 1 & 1 \end{array}\right]$$

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{ll} -3 & 4 \\ -1 & 1 \end{array}\right]$$

\(P\) is the transition matrix of a regular Markov chain. Find the long range transition matrix \(L\) of \(P\). $$P=\left[\begin{array}{ccc} 0.2 & 0.3 & 0.4 \\ 0.6 & 0.1 & 0.4 \\ 0.2 & 0.6 & 0.2 \end{array}\right]$$

Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and (d) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{llll} 2 & 1 & 1 & 0 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right]$$

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{lll} 3 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 3 \end{array}\right]$$

Use the power method to approximate the dominant eigenvalue and eigenvector of \(A\). Use the given initial vector \(\mathbf{x}_{0},\) the specified number of iterations \(k,\) and three-decimal-place accuracy. $$A=\left[\begin{array}{rr} -6 & 4 \\ 8 & -2 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right], k=6$$

Show that \(\lambda\) is an eigenvalue of \(A\) and find one eigenvector corresponding to this eigenvalue. $$A=\left[\begin{array}{ll} 4 & -2 \\ 5 & -7 \end{array}\right], \lambda=-6$$