Argue that if the matrix B is positive definite then so is t

Argue that if the matrix B is positive definite, then so is the matrix B^T.

Solution

Suppose B is positive definite then all eigen values of B are positive .

And since eigen values of the matrix and it\'s transpose are same

B\' also have only positive eigen values hence B\' is also positive definite.

For A and A\' have same eigen values,

Let c be eigen value of A then det(A-cI)=0

And since det A=det A\'

det(A-cI)=det(A-cI)\'=det(A\'-c\'I)

And det(A-cI)=0

Implies det(A\'-cI)=0

Implies that c is eigen value of A \'.