Find the characteristics roots and the corresponding vectors for the following matrix :
[ 1 0 -1
[ 1 2 1
[ 2 2 3]
Answers
Answer:
The eigen vector can be obtained from (A- λI)X = 0. Here A is the given matrix λ is a scalar,I is the unit matrix and X is the columns matrix formed by the variables a,b and c. Another name of characteristic Vector: Characteristic vector are also known as latent vectors or Eigen vectors of a matrix.
Step-by-step explanation:
think right
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 .
