This page covers the determination of eigenvalues and eigenspaces for matrices, focusing on triangular matrices and the characteristic polynomial, defined as the determinant of \(A - \lambda I_n\). It...This page covers the determination of eigenvalues and eigenspaces for matrices, focusing on triangular matrices and the characteristic polynomial, defined as the determinant of \(A - \lambda I_n\). It explains solving the characteristic polynomial to find eigenvalues, provides examples, and introduces the concept of eigenspaces. It discusses the trace of a matrix and offers recipes for calculating the characteristic polynomial, especially for 2x2 matrices.
An \(LU\) factorization of a matrix involves writing the given matrix as the product of a lower triangular matrix \(L\) which has the main diagonal consisting entirely of ones, and an upper triangular...An \(LU\) factorization of a matrix involves writing the given matrix as the product of a lower triangular matrix \(L\) which has the main diagonal consisting entirely of ones, and an upper triangular matrix \(U\) in the indicated order.
