Use Theorem 7.3 .4 to classify the matrix as positive...

use Theorem 7.3 .4 to classify the matrix as positive definite, negative definite, or indefinite.
(a) $A=\left[\begin{array}{rrr}3 & 1 & 2 \\ 1 & -1 & 3 \\ 2 & 3 & 2\end{array}\right]$
(b) $A=\left[\begin{array}{rrr}-3 & 2 & 0 \\ 2 & -3 & 0 \\ 0 & 0 & -5\end{array}\right]$

Key Concepts

Principal Minors

Principal minors are the determinants of the upper left k×k submatrices (for k=1,2,...,n) of a given matrix. They play a pivotal role in Sylvester’s Criterion to assess the definiteness of the matrix. The pattern of signs in these minors reveals whether the associated quadratic form is consistently of one sign or changes sign, thereby indicating if the matrix is positive definite, negative definite, or indefinite.

Definiteness of a Matrix

Definiteness refers to the sign behavior of the quadratic form x?Ax associated with a symmetric matrix A. A matrix is classified as positive definite if x?Ax > 0 for all nonzero x, negative definite if x?Ax < 0 for all nonzero x, and indefinite if x?Ax can take both positive and negative values. This concept is central in understanding stability in systems, optimization problems, and more.

Sylvester's Criterion

Sylvester's Criterion is a method used to determine the positive definiteness of a symmetric matrix by examining the signs of its leading principal minors. If all the leading principal minors are positive, then the matrix is positive definite. For negative definiteness, the signs of the minors must alternate starting with a negative value. This theorem provides a straightforward computational tool to classify matrices without directly computing eigenvalues.

Transcript

0:00 All right.

00:02 This problem is about classifying symmetric matrices as positive definite, negative definite, or indefinite.

00:08 And so what we need to do to figure that out is first we're going to compute the determinants of the principal sub -matrices.

00:17 And i will show you how to do that.

00:19 If they're all positive, then the matrix is positive definite.

00:23 If they're all negative, it's negative definite.

00:26 And if it's neither one of those, but it has at least one positive and one.

00:30 Negative determinant of a sub -matrix, then it's indefinite.

00:36 So let's take a look at what that means the determinant of the principal sub -matrices.

00:44 So here i have a three -by -three matrix.

00:49 My first sub -matracy is just the top -left corner.

00:54 Determinant of a one -by -one matrix is just the number itself.

00:58 So that's the determinant of my first sub -matrix.

01:03 Second sub -matrix is the top left corner with a two -by -two matrix.

01:10 So if i compute the determinant of that, i get three times negative 1, which is negative 3 minus 1, which is negative 4.

01:24 So i don't have all positive.

01:27 And then, sorry, back up.

01:28 The third thing i would need to do is i would need to compute the determinant of this matrix.

01:33 Okay, but i already know that not all of my determinants are positive.

01:42 They're not all negative.

01:44 I have one of each.

01:45 And so it can't be positive definite...

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