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