Textbook Guide Mathematics: Invertible Matrix, Gaussian Elimination, Coordinate Vector
Document Summary
Aim: using eigenvalues and eigenvectors in solving differential equation systems. find out if the equation has a nontrivial solution or not. For this, we need to follow the steps below: If (cid:1827)(cid:2206)=(cid:2019)(cid:2206) is provided, here (cid:2019) is a scalar value and is the eigenvalue if only the equation has a nontrivial solution, and (cid:2206) is the eigenvector related to the eigenvalue (cid:2019). If we are asked if a vector (cid:2206) is an eigenvector of a matrix (cid:1827) or not; we need to try (cid:1827)(cid:2206)=(cid:2019)(cid:2206) and find out if (cid:2206) satisfies the equation with any eigenvalue (cid:2019) or not. If it does, we say that vector (cid:2206) is an eigenvector of matrix (cid:1827). If we are asked if a scalar (cid:2019) is an eigenvalue of a matrix (cid:1827) or not; we first need to (cid:1827)(cid:2206)=(cid:2019)(cid:2206) (cid:1827)(cid:2206) (cid:2019)(cid:2206)=(cid:882) (cid:4666)(cid:1827) (cid:2019)(cid:4667)(cid:2206)=(cid:882) Here, if the columns of the matrix (cid:1827) (cid:2019) are linearly dependent, we say that the equation (cid:1827)(cid:2206)=(cid:2019)(cid:2206) has a nontrivial solution.
