Learning Objectives:

__Eigenvalues and Eigenvectors __

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Let **A** be an **n × n** matrix. Suppose is a non zero vector in **R ^{n}** and

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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