18.03 Lecture Notes - Lecture 21: Diagonalizable Matrix, Diagonal Matrix, Matrix Exponential
Document Summary
Apply diagonalization to an lti system to decouple it. Express the decoupled system in terms of the matrix exponential. Solve inhomogeneous lti systems using variation of parameters and matrix exponential. The matrix as maps to v , v. The matrix sd maps to v , v. Conclusion: as = sd so we get a = sds. The system x" = x is the same as x" = 3x and y" = 2y. This is a decoupled system, consisting of two odes that can be solved separately. More generally, if d is a diagonal matrix of any size, the inhomogeneous system x" = dx + q(t) consists of rst-order linear. Suppose that a is a 2 x 2 matrix with a basis of eigenvectors v , v having eigenvalues , . Using the eigenvalues to de ne a diagonal matrix d : = to v , v respectively is the matrix s : = whose columns are the eigenvectors.
