Let $A=(1-a)I_n + a J_n$
Find the values of $~a~$ so the Matrix is p.d ?
Note:$~I_n~$ is the identity matrix and $~J_n$ is the $1's$ matrix.
I know that $~~A$ is p.d $~iff~λ_i >0$ so, I need to choose $a$ so that $~λ_i >0$
First I tried to use the definition to find the eigen $det(A-λI_n)=0$
but it real complicated.
any help would be appreciated.
This question from A First Course in Linear Model Theory by Nalini, chapter 2.
Update: Trying to use the hint by @ whuber
By using the idea of inspection, I found the Gershgorin theorem ( first time I heard about it)
By the theorem we conclude that all$~ λi~$ in the disk
$D(1, (n-1)a)~~center~ at~ 1~, ~~and~ radius~ (n-1)a$
so in order to have positive value for $~ λi~$ we need to take
$|(n-1)a|<1~~$ hence, $-1/n-1<a<1/n-1$
@ whuber Is there another idea without using this theorem because we did not cover it in my class? Thanks