Problem: Find the conditions for $a$ such that matrix $A$ is positive definite.
$$A = \begin{pmatrix} 1 & a & a \\ a & 1 & a \\ a & a & 1 \\ \end{pmatrix} $$
Also, find the unitary matrix $U$ such that $U^{-1}AU$ is a diagonal matrix.
Attempt: I'm struggling to understand the definition and characteristics of positive definite. Can someone point me towards the solution?
one understanding of positive definitive is to have all eigenvalue positive,or whole determinant and also minor determinants should be greater then $0$,for example let us consider
$(1-a^2)>0$
which means that from negative infinity to $-1$ or from $1$ to infinity,also
$(a^2-a)>0$
or
$a(a-1)>0$
or from negative infinity to $0$ or $1$ to infinity
as determinant we have
syms a;
>> A=[1 a a;a 1 a;a a 1]
A =
[ 1, a, a]
[ a, 1, a]
[ a, a, 1]
>> det(A)
ans =
2*a^3 - 3*a^2 + 1
ans must be positive,or
[V D]=eig(A)
V =
[ -1, -1, 1]
[ 1, 0, 1]
[ 0, 1, 1]
D =
[ 1 - a, 0, 0]
[ 0, 1 - a, 0]
[ 0, 0, 2*a + 1]
we have
$(1-a)>0$ means that $1>a$
$2*a+1>0$
means that $a>-0.5$ ,so we have
$1>a$ and $a>-0.5$
about unitary matrix
