Eigenvalues & Eigenvectors

Eigenvalues & Eigenvectors

1. ·

   • Called an eigenvalue problem -> x is a column vector     “ Lamda” is a scalar -> values of lamda called eigenvalues
   • vector x is called an eigenvector of A corresponding to  .
   •  is called an eigenvalue of A if there is a nonzero vector x

2. Eigenspace

   • Definition
     • Let A be nxn matrix and let  be an eigenvalue of A. The collection of all eigenvectors corresponding to , together with zero vector
   • Eigenspace denoted by

3. Steps to find eigenvalue and eigenvector of a matrix

   • 1. Compute the characteristic polynomial det(A – I) of A. 2. Find the eigenvalues by solving the characteristic equations det(A–I)=0 for. 3. For each eigenvalue , find the space of the matrix A – I . This is the eigenspace E , the nonzero vectors of which are the eigenvectors of A corresponding to . 4. Find a basis for each eigenspace.

4. Example on how to solve the problem

   • here is the link on how to sole=ve the problem ; file:///C:/Users/User/Downloads/Scanned%20Documents%20(2).pdf

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