Find the eigen vectors and eigen values of the matrix.
A = [0,1][-2,-3]
Answers
Answer:
A=
⎣
⎢
⎢
⎡
2
2
−1
1
3
−1
1
4
−2
⎦
⎥
⎥
⎤
∣A−λI∣=0
⎣
⎢
⎢
⎡
2−λ
2
−1
1
3−λ
−1
1
4
−2−λ
⎦
⎥
⎥
⎤
=0
(2−λ)[(3−λ)(−2−λ)+4]−1[−4−2λ+4]+[−2+3−λ]=0
(2−λ)[λ
2
−λ−2]+λ+1=0
−λ
3
+3λ
2
+λ−3=0
λ
3
−3λ
2
−λ+3=0
(λ−1)(λ
2
−2λ−3)=0
(λ−1)(λ+1)(λ−3)=0
λ=−1,1,3
Therefore, Eigen values of matrix A are −1,1,3
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 .
