(a) Show that a symmetric matrix N is negative definite if...

(a) Show that a symmetric matrix $N$ is negative definite if and only if $K=-N$ is positive definite. (b) Write down two explicit criteria that tell whether a $2 \times 2$ matrix $N=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)$ is negative definite. (c) Use your criteria to check whether (i) $\left(\begin{array}{rr}-1 & 1 \\ 1 & -2\end{array}\right)$, (ii) $\left(\begin{array}{ll}-4 & -5 \\ -5 & -6\end{array}\right)$, (iii) $\left(\begin{array}{rr}-3 & -1 \\ -1 & 2\end{array}\right)$ are negative definite.

(a) Show that a symmetric matrix $N$ is negative definite if and only if $K=-N$ is positive definite. (b) Write down two explicit criteria that tell whether a $2 \times 2$ matrix $N=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)$ is negative definite.
(c) Use your criteria to check whether
(i) $\left(\begin{array}{rr}-1 & 1 \\ 1 & -2\end{array}\right)$,
(ii) $\left(\begin{array}{ll}-4 & -5 \\ -5 & -6\end{array}\right)$,
(iii) $\left(\begin{array}{rr}-3 & -1 \\ -1 & 2\end{array}\right)$ are negative definite.

Key Concepts

Principal Minor Criterion

Sylvester's criterion, or the principal minor criterion, is used to determine the positive definiteness of a matrix by checking if all leading principal minors are positive. For negative definiteness, one typically requires the principal minors’ signs to alternate in a specific pattern. In the case of 2×2 symmetric matrices, these conditions can be explicitly written in terms of the matrix entries, offering a practical method for verifying definiteness without computing eigenvalues.

Eigenvalue Criteria

The eigenvalue criterion is a key concept in assessing matrix definiteness. For symmetric matrices, being positive definite is equivalent to having all eigenvalues positive, and being negative definite corresponds to having all eigenvalues negative. This relationship provides a straightforward method for checking definiteness through spectral analysis, which is essential in many areas such as system stability and numerical methods.

Equivalence of Negative and Positive Definiteness

The principle that a symmetric matrix N is negative definite if and only if its negation -N is positive definite is an important conceptual insight. This equivalence comes from the fact that the quadratic form associated with -N is the negative of that associated with N. Thus, if all the quadratic evaluations of N are negative for nonzero vectors, then flipping the sign produces positive evaluations, a property characteristic of positive definiteness.

Definiteness of Matrices

Definiteness refers to whether a matrix, particularly in the context of quadratic forms, produces strictly positive, strictly negative, or indefinite outputs when evaluated with nonzero vectors. A matrix is positive definite if all nonzero vectors yield a positive scalar and negative definite if they yield a negative scalar. These properties are crucial in optimization, stability analysis, and various applications in linear algebra and statistics.

Symmetric Matrices

Symmetric matrices have the special property that they are equal to their transpose, which implies that all their eigenvalues are real. This attribute makes the analysis of definiteness more manageable since the sign of the eigenvalues directly corresponds to the behavior of the quadratic form. In many theoretical and practical contexts, symmetric matrices are preferred because of these well-behaved spectral properties.

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