## About "Characteristic vectors of matrix"

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

 1 1 3 1 5 1 3 1 1

Solution :

Let A =

 1 1 3 1 5 1 3 1 1

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

 1 0 0 0 1 0 0 0 1

Now we have to multiply λ with unit matrix I.

λI =

 λ 0 0 0 λ 0 0 0 λ

A-λI=

 1 1 3 1 5 1 3 1 1

-

 λ 0 0 0 λ 0 0 0 λ

=

 (1-λ) (1-0) (3-0) (1-0) (5-λ) (1-0) (3-0) (1-0) (1-λ)

=

 1-λ 1 3 1 5-λ 1 3 1 1-λ

|A-λI|=

 1-λ 1 3 1 5-λ 1 3 1 1-λ

= (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

=

 -2 1 3 1 2 1 3 1 -2

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 =

 1 -1 1

Substitute λ = -2 in the matrix A - λI

=

 3 1 3 1 7 1 3 1 3

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

 -1 0 1

Substitute λ = 6 in the matrix A - λI

=

 -5 1 3 1 -1 1 3 1 -5

The eigen vector z

 1 2 1

Let P =

 1 -1 1 -1 0 2 1 1 1

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

 3 0 0 0 -2 0 0 0 6

## Practice questions

(1)  Determine the characteristic vector of the matrix

 5 0 1 0 -2 0 1 0 5

(2)  Determine the characteristic vector of the matrix

 1 1 3 1 5 1 3 1 1

(3)  Determine the characteristic vector of the matrix

 -2 2 -3 2 1 -6 -1 -2 0

Solution

(4)  Determine the characteristic vector of the matrix

 4 -20 -10 -2 10 4 6 -30 -13

(5)  Determine the characteristic vector of the matrix

 11 -4 -7 7 -2 -5 10 -4 -6

## Related topics

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 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

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

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

1. Click on the HTML link code below.

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test