In this page characteristic vectors of matrix 2 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 2 :
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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= (1-λ)[ (5-λ)(1-λ) - 1 ] - 1[1 - λ - 3] + 3 [1 - 3 (5-λ) ] = (1-λ)[ 5 - 5 λ - λ + λ² - 1 ] - 1[ -λ - 2] + 3 [ 1 - 15 +3 λ ] = (1-λ)[ λ² - 6 λ + 4 ] + 1 λ + 2 + 3 [ - 14 +3 λ ] = λ² - 6 λ + 4 - λ³ + 6 λ² - 4 λ + λ + 2 - 42 + 9 λ = - λ³ + λ² + 6 λ² - 6 λ - 4 λ + λ + 9 λ + 4 + 2 - 42 = - λ³ + 7 λ² - 10 λ + 10 λ + 6 - 42 = - λ³ + 7 λ² - 36 = λ³ - 7 λ² + 36 |
To find roots let |A-λI| = 0
λ³ - 7 λ² + 36 = 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
λ² - 9 λ + 18 = 0
λ² - 3 λ - 6 λ + 18 = 0
λ (λ - 3) - 6 (λ - 3) = 0
(λ - 6) (λ - 3) = 0
λ - 6 = 0
λ = 6
λ - 3 = 0
λ = 3
Therefore the characteristic roots (or) Eigen values are x = -2,3,6
Substitute λ = -2 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.
3x + 1y + 3z = 0 ------ (1)
1x + 7y + 1z = 0 ------ (2)
3x + 1y + 3z = 0 ------ (3)
By solving (1) and (2) we get the eigen vector characteristic vectors of matrix2
The eigen vector x = |
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Substitute λ = 3 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.
-2x + 1y + 3z = 0 ------ (4)
1x + 2y + 1z = 0 ------ (5)
3x + 1y - 2z = 0 ------ (6)
By solving (4) and (5) we get the eigen vector
The eigen vector y = |
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Subsitute λ = 6 in the matrix A - λI characteristic vectors of matrix2
  = |
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From this matrix we are going to form three linear equations using variables x,y and z.
-5x + 1y + 3z = 0 ------ (7)
1x - 1y + 1z = 0 ------ (8)
3x + 1y - 5z = 0 ------ (9)
By solving (7) and (8) we get the eigen vector
The eigen vector z = |
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Questions |
Solution |
Question 1 : Determine the characteristic vector of the matrix
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characteristic vectors of matrix 2 | |||||||||||||
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 matrix 2 characteristic vectors of matrix 2 |
Mar 26, 23 08:27 PM
Mar 26, 23 08:26 PM
Mar 24, 23 05:25 AM