Eigenvalues & Eigenvectors - created from Mind Map

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Concepts

Eigenspaces

Finding Eigenvalues & Eigenvectors

Method to Obtain the Bases of Eigenspace A

Finding Eigenvalues of A^k

Eigenvalues are also called proper values, characteristic values, latent values

Eigenvector cannot be 0, but eigenvalue possible to be 0

if A is an n x n matrix with an eigenvalue and corresponding eigenvectors, then every nonzero scalar multiple of eigenvector is also an eigenvector of A

Also true that if eigenvectors are corresponding to the same eigenvalues, then their sum is also an eigenvector corresponding to eigenvalue

Set of all eigenvectors of a given eigenvalues, zero vectors, is the subspace of which is called the eigenspace of eigenvalue (lambda)

Let A be n x n matrix A

The determinant

Matrix A is singular iff lambda = 0 is an eigenvalue of A

The eigenvalues of A =[aij] are the roots of the characteristic polynomial of A

The eigenvalues problem in general case

The null space of is called the eigenspace of A corresponding to the eigenvalue lambda

Has nonzero solutions iff the coefficient matrix is not invertible ; det = 0

An eigenvalue of A is scalar lambda

Eigenvectors of A corresponding to lambda are nonzero solutions of :

characteristic equation of A

expanded ; characteristic polynomial of A

Characteristic polynomial of A

characteristic equation of A

since is nonzero, then it has a nontrivial solution if

Step 1 : FInd all eigenvalues of A

Step 2 : Find a basis for null space by solving the homogeneous system

Let lambda be an eigenvalue of A and x bar the corresponding eigenvector

A square matrix A is invertible iff lambda = 0 is not an eigenvalue of A

Assume that A is a square matrix with eigenvalues

Which means that lambda^3 is an eigenvalue of A^3 and x bar is the corresponding vector