Positive Definite Matrix

A Matrix ${\hbox{\sf A}}$ is positive definite if

$$({\hbox{\sf A}}{\bf v})\cdot {\bf v} > 0$$ (1)

for all Vectors ${\bf v} \not= 0$. 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 ${\hbox{\sf A}}$ is positive definite Iff there exists a Real nonsingular Matrix ${\hbox{\sf M}}$ such that

$${\hbox{\sf A}}={\hbox{\sf M}}{\hbox{\sf M}}^{\rm T}.$$ (2)

A $2\times 2$ Symmetric Matrix

$$\left[{\matrix{a & b\cr b & c\cr}}\right]$$ (3)

is positive definite if

$$a{v_1}^2+2b{v_1}{v_2}+c{v_2}^2 > 0$$ (4)

for all ${\bf v} = (v_1, v_2) \not = 0$.

A Hermitian Matrix ${\hbox{\sf A}}$ is positive definite if

1. $a_{ii}>0$ for all $i$,

2. $a_{ii}a_{ij}>\vert a_{ij}\vert^2$ for $i\not=j$,

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

4. $\vert{\hbox{\sf A}}\vert>0$.

See also Determinant, Eigenvalue, Hermitian Matrix, Matrix, Positive Semidefinite Matrix

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1106, 1979.