is Positive Definite Matrix Definition & Examples

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21. is Positive Definite Matrix example ( Enter your problem )

( Enter your problem )

20. is Periodic Matrix
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2. Example-2
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1. Definition & Examples

1. is Positive Definite Matrix ?
$[[25,15,-5],[15,18,0],[-5,0,11]]$

Solution:

$A$

=

	$25$	$15$	$-5$
	$15$	$18$	$0$
	$-5$	$0$	$11$

A matrix is positive definite if it's symmetric and all its pivots are positive.

Test method 1: Existence of all positive Pivots.
First apply Gaussian Elimination method to find Pivots

$A$

=

	$25$	$15$	$-5$
	$15$	$18$	$0$
	$-5$	$0$	$11$

$R_2 larr R_2-0.6xx R_1$

=

	$25$	$15$	$-5$
	$0$	$9$	$3$
	$-5$	$0$	$11$

$R_3 larr R_3+0.2xx R_1$

=

	$25$	$15$	$-5$
	$0$	$9$	$3$
	$0$	$3$	$10$

$R_3 larr R_3-0.3333xx R_2$

=

	$25$	$15$	$-5$
	$0$	$9$	$3$
	$0$	$0$	$9$

Pivots are the first non-zero element in each row of this eliminated matrix.

$:.$ Pivots are

$25,9,9$

Here all pivots are positive, so matrix is positive definite.

A matrix is positive definite if Determinants of all upper-left sub-matrices are positive.

Test method 2: Determinants of all upper-left sub-matrices are positive.

$A$

=

	$25$	$15$	$-5$
	$15$	$18$	$0$
	$-5$	$0$	$11$

$25$

$=25$

	$25$	$15$
	$15$	$18$

$=225$

	$25$	$15$	$-5$
	$15$	$18$	$0$
	$-5$	$0$	$11$

$=2025$

Determinants are

$25,225,2025$

Here all determinants are positive, so matrix is positive definite.

A matrix is positive definite if it's symmetric and all its eigenvalues are positive.

Test method 3: All positive eigen values.

$|A-lamdaI|=0$

	$(25-lamda)$	$15$	$-5$
	$15$	$(18-lamda)$	$0$
	$-5$	$0$	$(11-lamda)$

= 0

$:.(25-lamda)((18-lamda) × (11-lamda) - 0 × 0)-15(15 × (11-lamda) - 0 × (-5))+(-5)(15 × 0 - (18-lamda) × (-5))=0$

$:.(25-lamda)((198-29lamda+lamda^2)-0)-15((165-15lamda)-0)-5(0-(-90+5lamda))=0$

$:.(25-lamda)(198-29lamda+lamda^2)-15(165-15lamda)-5(90-5lamda)=0$

$:. (4950-923lamda+54lamda^2-lamda^3)-(2475-225lamda)-(450-25lamda)=0$

$:.(-lamda^3+54lamda^2-673lamda+2025)=0$

$:.-(lamda^3-54lamda^2+673lamda-2025)=0$

$:.(lamda^3-54lamda^2+673lamda-2025)=0$

Roots can be found using newton raphson method

Newton Raphson method for

$x^3-54x^2+673x-2025=0$

Here

$x^3-54x^2+673x-2025=0$

Let

$f(x) = x^3-54x^2+673x-2025$

$:. f'(x) = 3x^2-108x+673$

$x_0 = 4$

$1^(st)$ iteration :

$f(x_0)=f(4)=4^3-54 xx 4^2+673 xx 4-2025=-133$

$f'(x_0)=f'(4)=3 xx 4^2-108 xx 4+673=289$

$x_1 = x_0 - f(x_0)/(f'(x_0))$

$x_1=4 - (-133)/(289)$

$x_1=4.46020761$

$2^(nd)$ iteration :

$f(x_1)=f(4.46020761)=4.46020761^3-54 xx 4.46020761^2+673 xx 4.46020761-2025=-8.7977561$

$f'(x_1)=f'(4.46020761)=3 xx 4.46020761^2-108 xx 4.46020761+673=250.97793369$

$x_2 = x_1 - f(x_1)/(f'(x_1))$

$x_2=4.46020761 - (-8.7977561)/(250.97793369)$

$x_2=4.49526152$

$3^(rd)$ iteration :

$f(x_2)=f(4.49526152)=4.49526152^3-54 xx 4.49526152^2+673 xx 4.49526152-2025=-0.04986905$

$f'(x_2)=f'(4.49526152)=3 xx 4.49526152^2-108 xx 4.49526152+673=248.13388462$

$x_3 = x_2 - f(x_2)/(f'(x_2))$

$x_3=4.49526152 - (-0.04986905)/(248.13388462)$

$x_3=4.49546249$

$4^(th)$ iteration :

$f(x_3)=f(4.49546249)=4.49546249^3-54 xx 4.49546249^2+673 xx 4.49546249-2025=-0.00000164$

$f'(x_3)=f'(4.49546249)=3 xx 4.49546249^2-108 xx 4.49546249+673=248.11759994$

$x_4 = x_3 - f(x_3)/(f'(x_3))$

$x_4=4.49546249 - (-0.00000164)/(248.11759994)$

$x_4=4.4954625$

Approximate root of the equation

$x^3-54x^2+673x-2025=0$ using Newton Raphson method is

$4.4954625$ (After 4 iterations)

$n$	$x_0$	$f(x_0)$	$f'(x_0)$	$x_1$	Update
1	4	-133	289	4.46020761	$x_0 = x_1$
2	4.46020761	-8.7977561	250.97793369	4.49526152	$x_0 = x_1$
3	4.49526152	-0.04986905	248.13388462	4.49546249	$x_0 = x_1$
4	4.49546249	-0.00000164	248.11759994	4.4954625	$x_0 = x_1$

$:.$ x=4.4954625

Now, using long division

$(x^3-54x^2+673x-2025)/(x-4.4954625)=x^2-49.5045375x+450.45420817$

Now,

$x^2-49.5045375x+450.45420817=0$

$:. x=12.01568365$ and

$x=37.48885386$

$:.$ The eigenvalues of the matrix A are given by

$lamda=4.4954625,12.01568365,37.48885386$

Here all determinants are positive, so matrix is positive definite.

This material is intended as a summary. Use your textbook for detail explanation.
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20. is Periodic Matrix
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2. Example-2
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