is Negative Definite Matrix Example-2

2. is Negative Definite Matrix ?
[-100-1] $[[-1,0],[0,-1]]$

Solution:

A $A$

	-1 $-1$	0 $0$
	0 $0$	-1 $-1$

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

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

A $A$

	-1 $-1$	0 $0$
	0 $0$	-1 $-1$

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

∴ $:.$ Pivots are

-1,-1 $-1,-1$

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

A matrix is negative definite if Determinants

Di<0 $D_i<0$ for odd i and

Di>0 $D_i>0$ for even i .

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

A $A$

	-1 $-1$	0 $0$
	0 $0$	-1 $-1$

-1 $-1$

=-1 $=-1$

	-1 $-1$	0 $0$
	0 $0$	-1 $-1$

=1 $=1$

Determinants are

-1,1 $-1,1$

Here all odd determinants

D1<0 $D_1<0$ and even determinants

D2>0 $D_2>0$ , so matrix is negative definite.

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

Test method 3: All negative eigen values.

|A-λI|=0 $|A-lamdaI|=0$

	(-1-λ) $(-1-lamda)$	0 $0$
	0 $0$	(-1-λ) $(-1-lamda)$

= 0

∴(-1-λ)×(-1-λ)-0×0=0 $:.(-1-lamda) × (-1-lamda) - 0 × 0=0$

∴(1+2λ+λ2)-0=0 $:.(1+2lamda+lamda^2)-0=0$

∴(λ2+2λ+1)=0 $:.(lamda^2+2lamda+1)=0$

∴(λ+1)(λ+1)=0 $:.(lamda+1)(lamda+1)=0$

∴(λ+1)=0or(λ+1)=0 $:.(lamda+1)=0 or(lamda+1)=0$

∴ $:.$ The eigenvalues of the matrix

A $A$ are given by

λ=-1 $lamda=-1$ ,

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

This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here