use Theorem 7.3 .4 to classify the matrix as positive definite, negative definite, or indefinite

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

Key Concepts

Symmetric Matrices

A symmetric matrix is one that is equal to its transpose, meaning that its elements mirror each other across the main diagonal. This property is essential because many useful results in linear algebra, such as criteria for definiteness, are valid specifically for symmetric matrices.

Positive Definiteness

A symmetric matrix is called positive definite if its quadratic form produces a positive number for every nonzero vector. This is equivalent to the requirement that all eigenvalues of the matrix are positive, and it is also characterized by the fact that all leading principal minors (as stated in Sylvester's Criterion) are positive.

Negative Definiteness

A symmetric matrix is negative definite if its quadratic form yields a negative number for every nonzero vector. In terms of eigenvalues, this means they are all negative. Sylvester's Criterion characterizes this by requiring that the signs of the leading principal minors alternate, starting with a negative value.

Indefiniteness

A matrix is said to be indefinite if its quadratic form takes both positive and negative values depending on the input vector, which corresponds to the matrix having both positive and negative eigenvalues. In this case, the conditions for definiteness, such as those given by Sylvester's Criterion, are not satisfied.

Sylvester's Criterion

Sylvester's Criterion is a method used to determine the definiteness of a symmetric matrix by examining its leading principal minors. The matrix is positive definite if all these minors have positive determinants, and negative definite if they alternate in sign, starting with a negative determinant. This tool provides a straightforward way to classify the nature of a matrix without computing its eigenvalues directly.

Principal Minors

Principal minors of a matrix are the determinants of its submatrices obtained by deleting both corresponding rows and columns. In the context of definiteness, the leading principal minors – those formed by the first k rows and columns – are scrutinized to apply criteria like Sylvester's, offering insight into the nature of the quadratic form associated with the matrix.

Transcript

00:01 Okay, again, in this problem, we are classifying symmetric matrices as positive definite or negative definite or indefinite.

00:09 And so in this key idea here, i have the steps for doing that.

00:13 First, we have to compute the determinants of the principal sub -matrices, and i will show you how to do that in a second.

00:21 And then depending on whether those determinants are positive or negative, it's going to allow us to classify the matrix.

00:27 Okay.

00:29 So let's start here.

00:30 Let's compute the determinants of the principal matrix, submatrices for this example.

00:36 The first sub -matrix is just the first entry, the top left corner, the one -by -one matrix.

00:43 Okay? determinant of a one -by -one matrix is four.

00:47 I'm actually going to move that down here a little bit.

00:51 So that's the determinant of my first sub -matrix.

00:55 The determinant of my second sub -matrix is just the top.

00:59 Left 2 by 2 matrix so that's going to be 8 minus 1 which is 7 so just a quick note i don't know yet i can't classify it yet because i don't know if that third determinant which is going to be the determinant of all of this i don't know if that's positive or negative so i don't know if it's a positive definite or an indefinite yet so i'm gonna have to compute this 3 by 3 matrix here find it's determinant.

01:33 You guys know a few ways of doing that.

01:36 How i'm going to do it without technology.

01:39 So i'm going to do it this way.

01:44 This is a fun way that i learned how to do it.

01:47 So we're just going to copy down the first two columns.

01:50 I'm going to multiply these guys together.

01:52 Four times two times two is 16.

01:58 One times one times negative one is negative one.

02:02 And then negative one times 1 times 1 is negative 1.

02:08 So 16 minus 1 minus 1 gets me 14.

02:13 Then i'm going to go back the other way.

02:15 So 1 times 1 times 2 is 2.