Determine all possible real a, so that matrix #((1,a,a),(a,1,a),(a,a,1))# has 3 positive real eigen value ?

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Answer 1 · 2017-04-10T17:57:00

See below.

Giving #M = ((1,a,a),(a,1,a),(a,a,1))# it's characteristic polynomial is

#p(lambda)=lambda^3-3lambda^2+3(1-a^2)lambda-2a^3+3a^2-1 =#

#(lambda+a-1)^2(1+2a-lambda)#

so this polynomial has three real roots or

#p(lambda)=(lambda+a-1)^2(1+2a-lambda)#

with eigenvalues

#(1-a,1-a,1+2a)#.

This result does not surprise us because #M# is hermitian and a hermitian matrix has always real eigenvalues.

NOTE:
Positive eigenvalues are obtained for #1-a > 0# and #1+2a > 0# or

#-1/2 < a < 1#