Characteristic equation of A
det(λI - A) = 0
Theorem 5.1.1 Eigenvalue & characteristic equation
λ is an eigenvalue of A iff:
det(λI - A) = 0
Theorem 5.1.2:
Eigenvalues of an nxn triangular matrix:
Are the entries on the main diagonal.
Theorem 5.1.4 on powers of A and λ
If k is a positive integer, λ is an eigenvalue of matrix A, and x is a corresponding eigenvector, then λ^k is an eigenvalue of A^k and x is a corresponding eigenvector.
Theorem 5.1.5 on invertibility of square matrices and λ
A square matrix A is invertible iff λ = 0 is not an eigenvalue of A.
Similar matrices
If A and B are square matrices, then we say that B is similar to A if there is an invertible matrix P such that B = P^-1AP.
Diagonalisable matrix
A square matrix A is said to be diagonalisable if it is similar to some diagonal matrix.
If there exists some invertible matrix P such that P^-1AP is diagonal.
P is said to diagonalise A.
Theorem 5.2.1:
Equivalent statements on diagonalizability and eigenvectors.
If A is an n x n matrix:
- A is diagonalisable
- A has n linearly independent eigenvectors
Procedure for Diagonalizing a Matrix
- Confirm that the matrix is actually diagonalizable by finding n linearly independent eigenvectors. One way to do this is by finding a basis for each eigenspace and merging these basis vectors into a single set S. If this set has fewer than n vectors, then the matrix is not diagonalizable.
- Form the matrix P = [p1 p2 … pn] that has the vectors in S as its column vectors.
- The matrix P^-1AP will be the diagonal and have the eigenvalues λ1, λ2, … λn corresponding to the eigenvectors p1, p2, …, pn as its successive diagonal entries.
Theorem 5.2.2 on eigenvectors and linear independence
If v1, v2, …, vk are eigenvectors of a matrix corresponding to distinct eigenvalues, then {v1, v2, …, vk} is a linearly independent set.
Theorem 5.2.3 on diagonalisability and distinct eigenvectors
If an n x n matrix has n distinct eigenvalues, then A is diagonalizable.
Similarity invariant between A a
A
