Characteristic vectors of matrix :
Here we are going to see how to find characteristic equation of any matrix with example problems.
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 names of characteristic Vectors of matrix :
Characteristic vector are also known as latent vectors or Eigen vectors of a matrix.
Example :
Determine the characteristic vector of the matrix

Solution :
Let A = 

The order of A is 3 x 3. So the unit matrix I = 

Now we have to multiply λ with unit matrix I.
λI = 

AλI= 

 

= 

= 

AλI= 
 
= (1λ) [(5λ) (1λ)  1]1 [1(1λ)  3] + 3 [1  3 (5λ)] = (1λ) [55λλ+λ²1]  1 [1 λ  3] + 3[115+3λ] = (1λ) [λ²6λ+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 AλI = 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 (λ6) + 2 (λ6) = 0
(λ+2) (λ6) = 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
= 

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
The eigen vector x = 

Substitute λ = 2 in the matrix A  λI
= 

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 

Substitute λ = 6 in the matrix A  λI
= 

The eigen vector z 

Let P = 

The column of P are linearly independent eigen vectors of A . Therefore the diagonal matrix = 

(1) Determine the characteristic vector of the matrix

(2) Determine the characteristic vector of the matrix

(3) Determine the characteristic vector of the matrix

(4) Determine the characteristic vector of the matrix

(5) Determine the characteristic vector of the matrix

After having gone through the stuff given above, we hope that the students would have understood "Characteristic vectors of matrix"
Apart from the stuff given above, if you want to know more about "Characteristic vectors of matrix", please click here.
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits