Question

a and b

Using definitions, check whether the following matrices are positive definite or positive semidefinite: 1 . 1 (2) 4-61. 8-6]. c-[--B 11. --G 9. -- 1 2 0 0 (b) A= 0 1 0 0 0 1 2 37 2 4 6 3 6 0 B= -1 2 D= -1 2 -1 9 -4 2 1 -1

Answer 1

A = l'o clearly eigen-values of taxe = 11 Since both eigen' value cite the and non Zedo thin A is tve definite Matrix B Eigen-valies are: o.1 this both eigen- values are nonnigative then B is semipositive definite cois one of the eigen value.). C (9 It's Chasuiteastic <?-= a = II thus eigen-valus axe -1 and A so s is indefinite! 0-13 0 l ! since Dis Upped triangulus matsix so eigen values are it's element ie A and i are the eigen values of 0.80 Dis the definite matrix. main diagoncel

E = Ⓡ . clearly buth eigen-valus are so semi-definite o.and.o © A = u 0 Mi =10 ľ il Ma = 0 6 My - M.. Me and My be first second and this Minnot of matrix A. M,=o=Maant so A is kas indefinite. بیاد \ Mzco so e B 3 11 २ 6 3 to M. = lilal Ma = là al - M₂ - ܢ ܢܢ 2 3 G 3(12-12)-6 (6-6)=0 .3 G thus B ig semi-bolitive definite.

Ce:? M, T = Ma - la in M? 1 C (1) + (0-1)=1-1=0 cleadly mi=iso. Ma also Mi=130, Maal so and Maco 80 c is semi-positive definite. -1 2 ·DS 4 3 ri २ Mi All =-1 Ma = 12 21 = 9 - 9 = o 2 M₂ = 2 =-(0-20) + (100 2 thus Dis negative semi-definite.