solution of equation ax + xb = c by inversion of an m × m or n × n matrix
... for X, where X and C are M × N real matrices, A is an M × M real matrix, and B is an N × N real matrix. A familiar example occurs in the Lyapunov theory of stability [1], [2], [3] with B = AT . Is also arises in the theory of structures [4]. Using the notation P × Q to denote the Kronecker product ( ...
... for X, where X and C are M × N real matrices, A is an M × M real matrix, and B is an N × N real matrix. A familiar example occurs in the Lyapunov theory of stability [1], [2], [3] with B = AT . Is also arises in the theory of structures [4]. Using the notation P × Q to denote the Kronecker product ( ...
Exam 3 Sol
... (b) For each eigenvalue that you found in part (a), find a basis for the corresponding eigenspace. (c) Find the general solution of ~x0 = A~x. (Hint: you need to find a generalized eigenvector) Solution: (a) The characteristic polynomial of A is p(λ) = (1 − λ)(2 − λ)2 . The eigenvalues are λ1 = 1, λ ...
... (b) For each eigenvalue that you found in part (a), find a basis for the corresponding eigenspace. (c) Find the general solution of ~x0 = A~x. (Hint: you need to find a generalized eigenvector) Solution: (a) The characteristic polynomial of A is p(λ) = (1 − λ)(2 − λ)2 . The eigenvalues are λ1 = 1, λ ...
Solutions of First Order Linear Systems
... (c) Repeated Eigenvalues: If an eigenvalue is repeated we need to analyse the matrix A more carefully to find the corresponding vector solutions. Definition 1. The Algebraic Multiplicity (AM) of an eigenvalue λ is the number of times it appears as a root of the characteristic equation det(A − λI) = ...
... (c) Repeated Eigenvalues: If an eigenvalue is repeated we need to analyse the matrix A more carefully to find the corresponding vector solutions. Definition 1. The Algebraic Multiplicity (AM) of an eigenvalue λ is the number of times it appears as a root of the characteristic equation det(A − λI) = ...
Eigenvalues and Eigenvectors
... EIGENVECTORS AND EIGENVALUES The scalar λ is an eigenvalue of A if and only if the equation ( A λI )x 0 has a nontrivial solution, that is, if and only if the equation has a free variable. Because of the zero entries in A λI , it is easy to see that ( A λI )x 0 has a free variable if ...
... EIGENVECTORS AND EIGENVALUES The scalar λ is an eigenvalue of A if and only if the equation ( A λI )x 0 has a nontrivial solution, that is, if and only if the equation has a free variable. Because of the zero entries in A λI , it is easy to see that ( A λI )x 0 has a free variable if ...
Problem 1
... 2. Consider the minor cofactor expansion of det(A − λI) which gives a sum of terms. Each term is a product of n factors comprising one entry from each row and each column. Consider the minor cofactor term containing members of the diagonal (a11 − P λ)(a22 − λ) · · · (ann − λ). The coefficient for th ...
... 2. Consider the minor cofactor expansion of det(A − λI) which gives a sum of terms. Each term is a product of n factors comprising one entry from each row and each column. Consider the minor cofactor term containing members of the diagonal (a11 − P λ)(a22 − λ) · · · (ann − λ). The coefficient for th ...
