Let Aalpha beta alpha beta beta alpha where alpha and beta

Let A(alpha, beta) = (alpha beta beta alpha) where alpha and beta are parameters. What should be the relationships between alpha and beta for A(alpha, beta) to be positive definite, positive semidefinite, negative definite, negative semidefinite, neither of these?

Solution

Eigen values are (alpha - lambda)^2 - beta ^2 = 0

So lambda = alpha +/- beta

So if alpha > beta and then it is a positive definite matrix and alpha + beta >0

2)semidefinite matrix says A(Transpose) * M * A is positive or zero as well

A condition is all eigen values are positive and minors are positive. So Both alpha and beta are positve

Negative definite and negative semidefinite are exact opposites. eigens less than zeroe

and for semidefinite all minors are less than zero

So alhpha +/- beta < 0 for negative definite

and semidefinite alpha, beta less than zero

Other matrices are indefinite matrices