Find the Eigen values and Eigen vectors of matrix A. If A =[2 -1. 1
1. 2. -1
1. -1 2]
Answers
Concept of eigen values and eigen vectors :
♦ Let A be a square matrix and X be a non zero vector . Let λ be any scalar such that AX = λX . Then λ is called the eigen value (or characteristic root) and X is called the eigen vector (or characteristic vector) of the square matrix A .
♦ By definition , AX = λX
→ AX - λX = O , where O is the zero matrix of the order same as that of square matrix A .
→ (A - λɪ)X = O , where ɪ is the identity matrix of the same order as that of square matrix A .
→ BX = O , where B = A - λɪ
→ X = OB⁻¹
If B⁻¹ exists then X = O , but X ≠ O thus B⁻¹ doesn't exist .
If B⁻¹ doesn't exist then B must be a singular matrix .
→ |B| = 0
→ |A - λɪ| = 0 , which is called the characteristic equation of matrix A .
♦ A square matrix of order n×n has n eigen values . It may have repeated eigen values .
♦ Eigen vectors corresponding to distinct eigenvalues are linearly independent .
♦ Collection of all the eigen vectors of a square matrix A is called its eigen space .
