Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Learning Objectives:

Understand Eigenvalues and Eigenvectors.

Eigenvalues and Eigenvectors

 

Let A be an n × n matrix. Suppose  is a non zero vector in Rn and λ is a number or scalar such that  A.

This means A is a scalar multiple of .  is called an Eigenvector of A. λ is called an Eigenvalue of A.

We say λ is the Eigenvalue associated with or corresponding to .  is the Eigenvector associated with or corresponding to λ.

The word “Eigen” is German meaning ‘Proper’. Eigenvalues can also be called proper values, characteristics values, or latent roots.

The number λ is an Eigenvalue of A if and only if

                          or 

When expanded  is a polynomial in λ of degree n that is called the Characteristic Polynomial of A. The equation   is called the Characteristic equation.

Some points to remember:

1.    An Eigenvalue of A is a scalar λ, so that .

2.    The Eigenvectors of A corresponding to λ are the nonzero solutions of

 

Hook Questions:

1.      Give the definition of Eigenvalue and Eigenvector?

2.      What is “Eigen” means in German language?

3.      What is called Latent Roots?

4.      What is Characteristic Polynomial and Characteristic Equation

 

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