## Positive Definite Matrix

A Matrix is positive definite if

 (1)

for all Vectors . All Eigenvalues of a positive definite matrix are Positive (or, equivalently, the Determinants associated with all upper-left Submatrices are Positive).

The Determinant of a positive definite matrix is Positive, but the converse is not necessarily true (i.e., a matrix with a Positive Determinant is not necessarily positive definite).

A Real Symmetric Matrix is positive definite Iff there exists a Real nonsingular Matrix such that

 (2)

A Symmetric Matrix
 (3)

is positive definite if
 (4)

for all .

A Hermitian Matrix is positive definite if

1. for all ,

2. for ,

3. The element of largest modulus must lie on the leading diagonal,

4. .