In this page characteristic vectors of matrix1 we are going to see how to find characteristic equation of any matrix with detailed example.
Definition :
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.
Question 1 :
Determine the characteristic vector of the matrix
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To find eigen vector first let us find characteristic roots of the given matrix.
Let A = |
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The order of A is 3 x 3. So the unit matrix I = |
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Now we have to multiply λ with unit matrix I.
λI = |
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A-λI= |
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- |
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  = |
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  = |
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|A-λI|= |
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= (5-λ)[(-2-λ) (5-λ) - 0] - 0 [(0 - 0)] + 1 [0- (-2 -λ)] = (5-λ)[ -10 + 2 λ - 5 λ + λ²] - 0 + 2 + λ = (5-λ)[ -10 - 3 λ + λ²] - 0 + 2 + λ = (5-λ)[λ² -3 λ-10] + 2 + λ = 5 λ² - 15 λ - 50 - λ³ + 3 λ² + 10 λ + 2 + λ = - λ³ + 5 λ² + 3 λ² - 15 λ + 10 λ + λ - 50 + 2 = - λ³ + 8 λ² - 4 λ - 48 = λ³ - 8 λ² + 4 λ + 48 |
To find roots let |A-λI| = 0
λ³ - 8 λ² + 4 λ + 48 = 0
For solving this equation first let us do synthetic division.
By using synthetic division we have found one value of λ that is λ = -2.
Now we have to solve λ² - 10 λ + 24 to get another two values. For that let us factorize diagonalization of matrix1
λ² - 10 λ + 24 = 0
λ² - 6 λ - 4 λ + 24 = 0
λ (λ - 6) - 4 (λ - 6) = 0
(λ - 6) (λ - 4) = 0
λ - 6 = 0
λ = 6
λ - 4 = 0
λ = 4
Therefore the characteristic roots (or) Eigen values are x = -2,4,6
Substitute λ = -2 in the matrix A - λI characteristic vectors of matrix1
                = |
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From this matrix we are going to form three linear equations using variables x,y and z.
7x + 0y + 1z = 0 ------ (1)
0x + 0y + 0z = 0 ------ (2)
1x + 0y + 7z = 0 ------ (3)
By solving (1) and (3) we get the eigen vector characteristic vectors of matrix1
The eigen vector x = |
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Substitute λ = 4 in the matrix A - λI
                = |
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From this matrix we are going to form three linear equations using variables x,y and z.
1x + 0y + 1z = 0 ------ (4)
0x - 6y + 0z = 0 ------ (5)
1x + 0y + 4z = 0 ------ (6)
By solving (4) and (5) we get the eigen vector
The eigen vector y = |
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Substitute λ = 6 in the matrix A - λI characteristic vectors of matrix1
  = |
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From this matrix we are going to form three linear equations using variables x,y and z.
-1x + 0y + 1z = 0 ------ (7)
0x + 8y + 0z = 0 ------ (8)
1x + 0y - 1z = 0 ------ (9)
By solving (7) and (8) we get the eigen vector
The eigen vector z = |
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Questions |
Solution |
Question 2 : Determine the characteristic vector of the matrix
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Question 3 : Determine the characteristic vector of the matrix
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Question 4 : Determine the characteristic vector of the matrix
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Question 5 : Determine the characteristic vector of the matrix
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characteristic vectors of matrix1characteristic vectors of matrix 1 |
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