Definite Matrix

• Let be a self-adjoint operator on an -dimensional inner product space and where is an orthonormal for . Then is positive definite if .

  is positive semidefinite if

  A similar definition applies for matrices.

  • In other words, A positive definite matrix is one that defines an inner product. In particular we have the Quadratic Form

  • For positive semidefinite matrices, the criteria is weakened to be

• (Friedberg e6.4.13a) is positive definite if and only if all the eigenvalues are positive

  is positive semidefinite if and only if all the eigenvalues are nonnegative.

• (Friedberg e6.4.13b) is positive definite if and only if is also.

  is positive semidefinite if and only if is also.

• (Friedberg e6.4.13c) is positive definite if and only if

  For all nonzero -tuples .

• (Friedberg e6.4.13d) is positive semidefinite if and only if is a Gramian matrix.

• (Friedberg e6.4.16) Let and be positive definite on an inner product space .

  • is positive definite.
  • If , then is positive definite.
  • is positive definite.

Links §

• Friedberg, Insel and Spence