Determine whether the following symmetric matrices are...

Determine whether the following symmetric matrices are positive definite by computing their eigenvalues. Validate your conclusions by using the methods from Chapter 5. (a) $\left(\begin{array}{rr}2 & -2 \\ -2 & 3\end{array}\right)$ (b) $\left(\begin{array}{rr}-2 & 3 \\ 3 & 6\end{array}\right)$, (c) $\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right)$, (d) $\left(\begin{array}{rrr}4 & -1 & -2 \\ -1 & 4 & -1 \\ -2 & -1 & 4\end{array}\right)$.

Determine whether the following symmetric matrices are positive definite by computing their eigenvalues. Validate your conclusions by using the methods from Chapter 5.
(a) $\left(\begin{array}{rr}2 & -2 \\ -2 & 3\end{array}\right)$
(b) $\left(\begin{array}{rr}-2 & 3 \\ 3 & 6\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}4 & -1 & -2 \\ -1 & 4 & -1 \\ -2 & -1 & 4\end{array}\right)$.

Key Concepts

Eigenvalues

Eigenvalues are scalars associated with a matrix that satisfy the equation Av = ?v, where v is a nonzero eigenvector. In the context of testing for positive definiteness, the sign of the eigenvalues is used; a symmetric matrix is positive definite if and only if all its eigenvalues are positive.

Symmetric Matrices

A symmetric matrix is one that equals its own transpose, i.e., the element in the ith row and jth column is the same as that in the jth row and i-th column. This property ensures that all its eigenvalues are real and that it possesses a full set of orthogonal eigenvectors, which greatly simplifies analysis and computations such as checking for positive definiteness.

Positive Definite Matrices

A matrix is considered positive definite if, for every nonzero vector x, the quadratic form x?Ax is strictly positive. For symmetric matrices, this condition is equivalent to having all eigenvalues strictly positive. Positive definiteness is a critical concept in optimization and stability analysis.

Sylvester's Criterion

Sylvester's Criterion provides an alternative method to assess the positive definiteness of a symmetric matrix by requiring that all the leading principal minors (i.e., the determinants of the top-left k×k submatrices) are positive. This method circumvents the need to compute all eigenvalues directly while still giving a definitive check for positive definiteness.

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