Question

PROBLEM 5: Give a quick reason why each of these statements is true: 1. Every positive definite matrix is invertible. e only positive definite projection matrix is P_ 3. A diagonal matrix with positive diagonal entries is positive definite. 4. A symmetric matrix with a positive determinant might not be positive definite!

Answer 1

Every positive definite matrix is invertible If an n times n symmetric A is positive definite them all of its eigen values are positive, so 0 is not an eigen value of A therefore the system of equation A x = 0 has no non - trivial solution, and so A is invertible Or We can say, the determinant is positive as all eigen values are positive. The only positive definite projection matrix is P = I Rightarrow As we know, all the projection matrices except I are singular. That's why, the only positive definite projection matrix is P = I A diagonal matrix with positive diagonal entries is positive definite. Rightarrow In a diagonal matrix, the diagonal entries are its eigen values, and we know that for an symmetric matrix (positive definite) all of its eigen values are positive. Therefore a diagonal matrix with positive diagonal entries is