17 EXERCISES

17.1. Matrix products

Let f: \mathbb{R}^m \rightarrow \mathbb{R}^k and g: \mathbb{R}^n \rightarrow \mathbb{R}^m be two maps. Let h: \mathbb{R}^n \rightarrow \mathbb{R}^k be the composite map h=f \circ g, with values h(x)=f(g(x)) for x \in \mathbb{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).

17.2 Special matrices

A matrix P \in \mathbb{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.

17.3. Linear maps, dynamical systems

1. Let f: \mathbb{R}^n \rightarrow \mathbb{R}^m be a linear map. Show how to compute the (unique) matrix A such that f(x)=A x for every x \in \mathbb{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 \mathbb{R}^n, input vector u \in \mathbb{R}^p, and output vector y \in \mathbb{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 \mathbb{R}^{n \times n}, \; B \in \mathbb{R}^{n \times p},  and C \in \mathbb{R}^{k \times n} given matrices.

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 ?

17.4 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 \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times p}, and let C:=A B \in \mathbb{R}^{m \times p}. Show that \|C\| \leq\|A\| \cdot\|B\| where \|\cdot\| denotes the largest singular value norm of its matrix argument.

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