Exercises

L. El Ghaoui

Exercises

Matrix products

1. Let $f: \mathbf{R}^m \rightarrow \mathbf{R}^k$ and $g: \mathbf{R}^n \rightarrow \mathbf{R}^m$ be two maps. Let $h: \mathbf{R}^n \rightarrow \mathbf{R}^k$ be the composite map $h=f \circ g$ , with values $h(x)=f(g(x))$ for $x \in \mathbf{R}^n$ . Show that the derivatives of $h$ can be expressed via a matrix-matrix product, as $J_h(x)=J_f(g(x)) \cdot J_g(x)$ , where the Jacobian matrix of $h$ at $x$ is defined as the matrix $J_h(x)$ with $(i, j)$ element $\partial h_i / \partial x_j(x)$ .

Special matrices

1. A matrix $P \in \mathbf{R}^{n \times n}$ is a permutation matrix if it is a permutation of the columns of the $n \times n$ identity matrix.
a. For a $n \times n$ matrix $A$ , we consider the products $P A$ and $A P$ . Describe in simple terms what these matrices look like with respect to the original matrix $A$ .
b. Show that $P$ is orthogonal.
c. Show that $P^2=I$ .

Linear maps, dynamical systems

1. Let $f: \mathbf{R}^n \rightarrow \mathbf{R}^m$ be a linear map. Show how to compute the (unique) matrix $A$ such that $f(x)=A x$ for every $x \in \mathbf{R}^n$ , in terms of the values of $f$ at appropriate vectors, which you will determine.
2. Consider a discrete-time linear dynamical system (for background, see here) with state $x \in \mathbf{R}^n$ , input vector $u \in \mathbf{R}^p$ , and output vector $y \in \mathbf{R}^k$ , that is described by the linear equations $x(t+1)=A x(t)+B u(t), \quad y(t)=C x(t)$ ,
with $A \in \mathbf{R}^{n \times n}, B \in \mathbf{R}^{n \times p}$ , and $C \in \mathbf{R}^{k \times n}$ given matrices.
1. a. Assuming that the system has initial condition $x(0)=0$ , express the output vector at time $T$ as a linear function of $u(0), \ldots, u(T)$ ; that is, determine a matrix $H$ such that $y(T)=H \bar{u}(T)$ , where $\bar{u}(T):=(u(0), \ldots, u(T-1))$ is a vector containing all the inputs up to and including at time $T-1$ .
b. What is the interpretation of the range of $H$ ?

Matrix inverses, norms

1. Show that a square matrix is invertible if and only if its determinant is non-zero. You can use the fact that the determinant of a product is a product of the determinant, together with the QR decomposition of the matrix $A$ .
2. Let $A \in \mathbf{R}^{m \times n}, B \in \mathbf{R}^{n \times p}$ , and let $C:=A B \in \mathbf{R}^{m \times p}$ . Show that $\|C\| \leq\|A\| \cdot\|B\|$ where $\|\cdot\|$ denotes the largest singular value norm of its matrix argument.

Exercises

Matrix products

Special matrices

Linear maps, dynamical systems

Matrix inverses, norms

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