is Negative Definite Matrix Example-2

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

Solution:

`A`

	`-1`	`0`
	`0`	`-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`

	`-1`	`0`
	`0`	`-1`

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

`:.` Pivots are `-1,-1`

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

A matrix is negative definite if Determinants `D_i<0` for odd i and `D_i>0` for even i .

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

`A`

	`-1`	`0`
	`0`	`-1`

`-1`

`=-1`

	`-1`	`0`
	`0`	`-1`

`=1`

Determinants are `-1,1`

Here all odd determinants `D_1<0` and even determinants `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-lamdaI|=0`

	`(-1-lamda)`	`0`
	`0`	`(-1-lamda)`

= 0

`:.(-1-lamda) × (-1-lamda) - 0 × 0=0`

`:.(1+2lamda+lamda^2)-0=0`

`:.(lamda^2+2lamda+1)=0`

`:.(lamda+1)(lamda+1)=0`

`:.(lamda+1)=0 or(lamda+1)=0 `

`:.` The eigenvalues of the matrix `A` are given by `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.
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2. Example-2