91Ó°ÊÓ

Let \(A\) be an \(n \times n\) matrix and let \(B=A-\alpha I\) for some scalar \(\alpha .\) How do the eigenvalues of \(A\) and \(B\) compare? Explain.

Let \(A\) be a singular \(n \times n\) matrix. Show that \(A^{T} A\) is positive semidefinite, but not positive definite.

Let \(A\) be an \(n \times n\) matrix and let \(B=A+I\). Is it possible for \(A\) and \(B\) to be similar? Explain.

Let \(A\) be a symmetric positive definite matrix. Show that the diagonal elements of \(A\) must all be positive.

Let \(A\) be an \(n \times n\) matrix with an eigenvalue \(\lambda\) of multiplicity \(n .\) Show that \(A\) is diagonalizable if and only if \(A=\lambda I\)

Show that \(A\) and \(A^{T}\) have the same eigenvalues. Do they necessarily have the same eigenvectors? Explain.

Let \(A\) be a symmetric \(n \times n\) matrix. Show that \(e^{A}\) is symmetric and positive definite.

Let \(A\) be an \(n \times n\) positive stochastic matrix with dominant eigenvalue \(\lambda_{1}=1\) and linearly independent eigenvectors \(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n},\) and let \(\mathbf{y}_{0}\) be an initial probability vector for a Markov chain \\[ \mathbf{y}_{0}, \mathbf{y}_{1}=A \mathbf{y}_{0}, \mathbf{y}_{2}=A \mathbf{y}_{1}, \dots \\] (a) Show that \(\lambda_{1}=1\) has a positive eigenvector \(\mathbf{x}_{1}\) (b) Show that \(\left\|\mathbf{y}_{j}\right\|_{1}=1, j=0,1, \ldots\) (c) Show that if \\[ \mathbf{y}_{0}=c_{1} \mathbf{x}_{1}+c_{2} \mathbf{x}_{2}+\cdots+c_{n} \mathbf{x}_{n} \\] then the component \(c_{1}\) in the direction of the positive eigenvector \(\mathbf{x}_{1}\) must be nonzero. (d) Show that the state vectors \(\mathbf{y}_{j}\) of the Markov chain converge to a steady-state vector. (e) Show that \\[ c_{1}=\frac{1}{\left\|\mathbf{x}_{1}\right\|_{1}} \\] and hence the steady-state vector is independent of the initial probability vector \(\mathbf{y}_{0}\)

Let \(A\) be a diagonalizable matrix and let \(X\) be the diagonalizing matrix. Show that the column vectors of \(X\) that correspond to nonzero eigenvalues of \(A\) form a basis for \(R(A)\)

Let \(A\) be a \(2 \times 2\) matrix. If \(\operatorname{tr}(A)=8\) and \(\operatorname{det}(A)=\) \(12,\) what are the eigenvalues of \(A ?\)