Due on Wednesday, Jan 30.
Suppose that \(A \in M_n\) is invertible and \(\lambda \in \sigma(A)\). Show that \(\lambda^{-1} \in \sigma(A^{-1})\). Hint: How does \(A^{-1}\) transform an eigenvector of \(A\)?
If \(p(x) = a_k x^k + a_{k-1} x^{k-1} + \ldots +a_1 x + a_0\) is a polynomial with complex coefficients and \(A \in M_n\), then we define \(p(A)\) to be the matrix \(a_k A^k + a_{k-1} A^{k-1} + \ldots + a_1 A + a_0 I\). Show that the spectrum of the matrix \(p(A)\) is the set \(\{p(\lambda): \lambda \in \sigma(A)\}\). (This is a theorem in Horn & Johnson, but it is a good exercise to prove it yourself. To get started, take any eigenvector \(x\) of \(A\), and calculate \(p(A)x\).)
Show that the all ones vector \(e = [1, 1, \ldots, 1]^T\) is an eigenvector of a matrix \(A \in M_n\) with eigenvalue 1 if and only if the entries in every row of \(A\) add up to 1.
Show that if \(A \in M_n(\mathbb{R})\) has a real eigenvalue \(\lambda\), then \(A\) must also have a real eigenvector \(x\) corresponding to \(\lambda\). Hint: Suppose that \(A z = \lambda z\) where \(z \ne 0\) is a complex eigenvector. Show that \(z = x + i y\) where both \(x\) and \(y\) are real vectors. What can you say about \(x\)? What happens if \(x = 0\)?
A matrix \(A \in M_n\) is idempotent if \(A^2 = A\). Show that every eigenvalue of an idempotent matrix must be either 0 or 1.
If \(\lambda\) is an eigenvalue of \(A \in M_n\), show that there is a unit vector \(x\) such that \(\lambda = \langle Ax, x \rangle\). (Recall that a unit vector is a vector with norm equal to 1.)
A matrix \(A \in M_n\) is Hermitian if \(A^* = A\). Prove that all eigenvalues of a Hermitian matrix are real. Hint: Show that \(\langle Ax,x \rangle = \langle x, A^*x \rangle = \langle x, Ax \rangle\) and use the conjugate-symmetry property of the inner product.
Let \(x, y\) be nonzero vectors in \(\mathbb{C}^n\). Show that \(x\) is an eigenvector of the matrix \(xy^*\). What is its eigenvalue?