17 EXERCISES
17.1. Matrix products
Let and be two maps. Let be the composite map , with values for . Show that the derivatives of can be expressed via a matrix-matrix product, as , where the Jacobian matrix of at is defined as the matrix with element .
17.2 Special matrices
A matrix is a permutation matrix if it is a permutation of the columns of the identity matrix.
a. For a matrix , we consider the products and . Describe in simple terms what these matrices look like with respect to the original matrix .
b. Show that is orthogonal.
c. Show that .
17.3. Linear maps, dynamical systems
1. Let be a linear map. Show how to compute the (unique) matrix such that for every , in terms of the values of at appropriate vectors, which you will determine.
2. Consider a discrete-time linear dynamical system (for background, see here) with state , input vector , and output vector , that is described by the linear equations
with , and given matrices.
a. Assuming that the system has initial condition , express the output vector at time as a linear function of ; that is, determine a matrix such that , where is a vector containing all the inputs up to and including at time .
b. What is the interpretation of the range of ?
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 .
2. Let , and let . Show that where denotes the largest singular value norm of its matrix argument.