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Magazine Problem 4
Show that if \(z\) is a nonzero complex number, then there are exactly two complex numbers \(w\) such that \(w^{2}=z\) If \(z\) is in polar form, describe \(w\) in polar form.
Problem 4
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. $$\left[\begin{array}{rr} 0 & -1 \\ 1 & 2 \end{array}\right]$$
Problem 4
Let \(A\) be an invertible \(n \times n\) matrix and \(\vec{v}\) an eigenvector of \(A\) with associated eigenvalue \(\lambda\). Is \(\vec{v}\) an eigenvector of \(7 A ?\) If so, what is the eigenvalue?
Problem 4
For each of the matrices find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology. $$\left[\begin{array}{rr} 0 & 4 \\ -1 & 4 \end{array}\right]$$
Problem 5
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. $$\left[\begin{array}{rr} 4 & 5 \\ -2 & -2 \end{array}\right]$$
Problem 5
Show that if \(z\) is a nonzero complex number, then there exist exactly \(n\) complex numbers \(w\) such that \(w^{n}=z\) If \(z\) is in polar form, write \(w\) in polar form. Represent the vectors \(w\) in the complex plane.
Problem 5
For the matrices \(A\) in Exercises 1 through \(10,\) determine whether the zero state is a stable equilibrium of the dynamical system \(\vec{x}(t+1)=A \vec{x}(t)\). $$A=\left[\begin{array}{rr} 0.5 & 0.6 \\ -0.3 & 1.4 \end{array}\right]$$
Problem 5
For each of the matrices find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology. $$\left[\begin{array}{rr} 11 & -15 \\ 6 & -7 \end{array}\right]$$
Problem 6
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. $$\left[\begin{array}{ll} 2 & 3 \\ 4 & 5 \end{array}\right]$$
Problem 6
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.
