| # Exercises Matrices > Exercises ## 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 . ## Special matrices A matrix is a permutation matrix if it is a permutation of the columns of the identity matrix. For a matrix , we consider the products and . Describe in simple terms what these matrices look like with respect to the original matrix . Show that is orthogonal. Show that . ## Linear maps, dynamical systems 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. 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. 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 . What is the interpretation of the range of ? ## Matrix inverses, norms 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 . Let , , and let . Show that where denotes the largest singular value norm of its matrix argument.