Positive Definite Matrix

Positive definite matrices are said to be common in applications of Linear Algebra. They are characterized by a set of conditions that, if one is fulfilled all the other are also true. A matrix $A=\left[\begin{array}{ccccccc}2& -1& 0-1& 2& -1\text{ }0& -1& 2\end{array}\right]$ is p.d. iff these are true:

• All sub-determinants of $A>0$ ($1\times 1\dots n\times n$)
• Eigenvalues all $\lambda >0$
• All pivots $>0$
• $x^{T}Ax>0$ for all $\vec{x}\ne \vec{0}$

Polynomials from $x^{T}Ax$

p.d. matrices are brought into the form of an n-degree polynomial for any $x=\text{[}x\_1,\dots ,x\_n]$.

$$\text{Polynomial of A}=f(x\_1,x\_2,x\_3)=2x\_1^2+2x\_2^2+2x\_3^2-2x\_1x\_2-2x\_2x\_3$$

When $A$ is positive-definite $f(x\_1,x\_2,\dots ,x\_n)>0$ for all $x$.

Elliptic Cross-sections

Cutting a cross section from $f(x\_1,\dots ,x\_n)$ it is a $n$ dimensional representative out of the family of ellipses iff the matrix is positive definite. For our example of $3\times 3$ the cross-section is a paraboloid (olive-shaped), whereas $2\times 2$ yields a 2d-ellipse. The principal axes of the resulting geometry are the directions of the Eigenvectors and the lengths are given by Eigenvalues $\lambda$. For a circle, all $\lambda =\cdots =\lambda$ (all lengths of axes are equal)