Lecture
This lecture defines and elaborates on the concept of the exponential norm of a matrix, building on previous discussions. It begins with the definition of matrix norms, explaining how to calculate the norm of a square matrix by summing the squares of its coefficients and taking the square root. The instructor presents three main properties of matrix norms, including the relationship between the norm of a product of matrices and the norms of the individual matrices. Following this, the lecture transitions to the exponential of a matrix, defined using the exponential series. The instructor discusses the convergence of this series for any square matrix and outlines several important properties, such as the behavior of the exponential of the zero matrix and the identity matrix. The lecture concludes with a demonstration of the convergence of the exponential series, establishing that the partial sums form a Cauchy sequence, thereby proving the existence of the matrix exponential.