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=λ.
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
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
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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