Definite, Semi-Definite and Indefinite Matrices

Definite, Semi-Definite and Indefinite Matrices

We are about to look at an important type of matrix in multivariable calculus known as Hessian Matrices. We will then formulate a generalized second derivatives test for a real-valued function $z = f(x_1, x_2, ..., x_n)$ of $n$ variables with continuous partial derivatives at a critical point $\mathbf{a} = (a_1, a_2, ..., a_n) \in D(f)$ to determine whether $f(\mathbf{a})$ is a local maximum value, local minimum value, or saddle point of $f$.

Before we do this though, we will need to be able to analyze whether a square $n \times n$ symmetric matrix is positive definite, negative definite, indefinite, or positive/negative semidefinite. These terms are more properly defined in Linear Algebra and relate to what are known as eigenvalues of a matrix. We will now go into the specifics here, however, the definition below will be sufficient for what we need.

 Definition: Let $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}$ be an $n \times n$ symmetric matrix, and let $D_i = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1i} \\ a_{21} & a_{22} & \cdots & a_{2i} \\ \vdots & \vdots & \ddots & \vdots \\ a_{i1} & a_{i2} & \cdots & a_{ii} \end{vmatrix}$ for $i = 1, 2, ..., n$. Then: a) $A$ is said to be Positive Definite if $D_i > 0$ for $i = 1, 2, ..., n$. b) $A$ is said to be Negative Definite if $D_i < 0$ for odd $i \in \{ 1, 2, ..., n \}$ and $D_i > 0$ for even $i \in \{ 1, 2, ..., n \}$. c) $A$ is said to be Indefinite if $\det (A) = D_n \neq 0$ and neither a) nor b) hold. d) If $\det (A) = D_n = 0$, then $A$ may be Indefinite or what is known Positive Semidefinite or Negative Semidefinite.

The values $D_i$ for $i = 1, 2, ..., n$ are the values of the determinants of the $i \times i$ top left submatrices of $A$. Note that $D_1 = a_{11}$, $D_2 = \begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix}$, etc…

Let's look at some examples of classifying square symmetric matrices.

Example 1

Classify the following square symmetric matrix $A = \begin{bmatrix} 6 & 4\\ 4 & 5 \end{bmatrix}$ as positive definite, negative definite, indefinite, or unknown.

We have that $D_1 = 6 > 0$, and $D_2 = \begin{vmatrix} 6 & 4\\ 4 & 5 \end{vmatrix} = 30 - 16 = 14 > 0$. Therefore, $A$ is a positive definite matrix.

Example 2

Classify the following square symmetric matrix $\begin{bmatrix} -3 & 0 & 0\\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ as positive definite, negative definite, indefinite, or unknown.

We have that $D_1 = -3 < 0$ and $D_2 = \begin{vmatrix} -3 & 0\\ 0 & -2 \end{vmatrix} = 6 > 0$. The matrix $A$ is known as a diagonal matrix, and the determinant $D_3 = \begin{vmatrix} -3 & 0 & 0\\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{vmatrix}$ can be computed as the product of the entries in the main diagonal, that is $D_3 = (-3)(-2)(-1) = -6 < 0$.

Since $D_1, D_3 < 0$ and $D_2 > 0$, we have that $A$ is a negative definite matrix.