91Ó°ÊÓ

For the matrices \(A\) in Exercises 1 through \(10,\) determine whether the zero state is a stable equilibrium of the dynamical system \(\vec{x}(t+1)=A \vec{x}(t)\). $$A=\left[\begin{array}{rr} -1 & 3 \\ -1.2 & 2.6 \end{array}\right]$$

If \(z\) is a nonzero complex number in polar form, describe \(1 / z\) in polar form. What is the relationship between the complex conjugate \(\bar{z}\) and \(1 / z ?\) Represent the numbers \(z, \bar{z},\) and \(1 / z\) in the complex plane.

For each of the matrices \(A\) find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize \(A,\) if you can. Do not use technology. $$\left[\begin{array}{ll} 2 & 3 \\ 4 & 5 \end{array}\right]$$

Describe the transformation \(T(z)=(1-i) z\) from \(\mathbb{C}\) to \(\mathbb{C}\) geometrically.

If \(\vec{v}\) is an eigenvector of the \(n \times n\) matrix \(A\) with associated eigenvalue \(\lambda,\) what can you say about $$\operatorname{ker}\left(A-\lambda I_{n}\right) ?$$ Is the matrix \(A-\lambda I_{n}\) invertible?

For each of the matrices find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology. $$I_{3}$$

For each of the matrices \(A\) find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize \(A,\) if you can. Do not use technology. $$\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right]$$

For the matrices \(A\) in Exercises 1 through \(10,\) determine whether the zero state is a stable equilibrium of the dynamical system \(\vec{x}(t+1)=A \vec{x}(t)\). $$A=\left[\begin{array}{rr} 2.4 & -2.5 \\ 1 & -0.6 \end{array}\right]$$

For the matrices \(A\) in Exercises 1 through \(10,\) determine whether the zero state is a stable equilibrium of the dynamical system \(\vec{x}(t+1)=A \vec{x}(t)\). $$A=\left[\begin{array}{rr} 1 & -0.2 \\ 0.1 & 0.7 \end{array}\right]$$

For each of the matrices \(A\) find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize \(A,\) if you can. Do not use technology. $$\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{array}\right]$$