Question

find the eigen values and eigen vectors of the matrix A = [ 2 -2 3 , 1 1 1 , 1 3 -1 ]​

Answer 1

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 .