In this page characteristic vectors of matrix 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.
Example :
Determine the characteristic vector of the matrix
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Solution :
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(1-λ) - 3] + 3 [1 - 3 (5-λ)] = (1-λ) [5-5λ-λ+λ²-1] - 1 [1- λ - 3] + 3[1-15+3λ] = (1-λ) [λ²-6λ+4] - 1[-2 - λ] + 3[-14+3λ] = λ² - 6λ + 4 - λ³ + 6λ² - 4λ + 2 + λ - 42 + 9λ = - λ³ + 7 λ² - 10λ + 10λ + 6 - 42 = - 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 = 3.
Now we have to solve λ² - 4 λ - 12 to get another two values.For that let us use factoring method.characteristic vectors of matrix
λ² - 4 λ - 12 = 0
λ² - 6 λ + 2 λ - 12 = 0
λ (λ-6) + 2 (λ-6) = 0
(λ+2) (λ-6) = 0
λ + 2 = 0 λ - 6 = 0
λ = -2 λ = 6
Therefore the characteristic roots are x = 3,-2 and 6
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 ------ (1)
1x + 2y + 1z = 0 ------ (2)
3x + 1y - 2z = 0 ------ (3)
By solving (1) and (2) we get the eigen vector characteristic vectors of matrix characteristic vectors of matrix
The eigen vector x = |
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Substitute λ = -2 in the matrix A - λI characteristic vectors of matrix
                = |
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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 ------ (4)
1x + 7y + 1z = 0 ------ (5)
3x + 1y + 3z = 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
                = |
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The eigen vector z = |
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Let P = |
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The column of P are linearly independent eigen vectors of A . Therefore the diagonal matrix = |
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Questions |
Solution |
Question 1 : Determine the characteristic vector of the matrix
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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 matrix |
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