Julia: Positive and Negative Definite Matrices

Functions to create surfaces

These functions implement the expression, which can calculate scalar quantities when given x1 and x2 components of a 2-element vector.

\[ f(\mathbf x) = \mathbf x^T \mathbf A \mathbf x \]

In this case, I take the matrix A to positive definite or negative definite. Then, I pass the same vectors into the function to see the behavior of positive or negative definite matrices.

Positive definite

The 2x2 identity matrix is positive definite:

\[ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \]

function pos_definite(x1::Float64, x2::Float64)
    A = [1.0 0.0; 0.0 1.0]
    x = [x1; x2]
    x' * A * x
end

pos_definite (generic function with 1 method)

Negative definite

The following 2x2 matrix is negative definite:

\[ \begin{bmatrix} -1 & 0 \\ 0 & -2 \\ \end{bmatrix} \]

function neg_definite(x1::Float64, x2::Float64)
    A = [-1.0 0.0; 0.0 -2.0]
    x = [x1; x2]
    x' * A * x
end

neg_definite (generic function with 1 method)

Surface plots

Positive definite plot

xs = -5:0.1:5
ys = -5:0.1:5

zs = [pos_definite(x, y) for x in xs, y in ys]

display(surface(xs, ys, zs, title="Positive Definite Matrix Transform", xlabel="x", ylabel="y", zlabel="z", cmap=:cool))

Negative definite plot

xs = -5:0.1:5
ys = -5:0.1:5

zs = [neg_definite(x, y) for x in xs, y in ys]

display(surface(xs, ys, zs, title="Negative Definite Matrix Transform", xlabel="x", ylabel="y", zlabel="z", cmap=:cool))