express the quadratic form in the matrix notation 𝐱^T A 𝐱, where A is a symmetric matrix. (a) 5

Express the quadratic form in the matrix notation 𝐱^T A 𝐱,...

Key Concepts

Coefficient Assignment in Matrix Representation

When converting a quadratic form from its standard polynomial representation to matrix notation, one must match the coefficients of the squared terms and the cross-product terms to the entries of the matrix A. For the cross terms, the coefficients in the polynomial are distributed equally between the symmetric entries of A since each cross term appears twice in the expansion of x?Ax. This careful distribution is key to ensuring that the matrix accurately represents the quadratic form.

Quadratic Forms

A quadratic form is a homogeneous polynomial of degree two in a number of variables. It can be expressed in a general form as a sum of squared terms and cross-product terms. This concept is fundamental in linear algebra because quadratic forms appear in optimization, statistics (e.g., covariance matrices), and many areas of applied mathematics.

Matrix Notation of Quadratic Forms

The matrix notation for a quadratic form utilizes the expression x?Ax, where x is a column vector of variables and A is a matrix whose entries are the coefficients related to the variables in the quadratic form. This compact representation is particularly useful for theoretical analysis and computational implementation, as it facilitates the use of eigenvalue methods and other matrix operations.

Symmetric Matrices

In the context of quadratic forms, the matrix A is usually taken to be symmetric. This is because any quadratic form has an associated symmetric matrix representation, with the off?diagonal elements accounting for the cross-terms. The symmetry ensures that the form x?Ax is real for any vector x and also simplifies many proofs and computations in linear algebra.

Transcript

00:01 All right, the key idea for this problem where we want to take an expression for a quadratic form that looks like an algebra expression and translate it to a matrix expression is knowing where we get the entries in our matrix from.

00:16 And so we're going to demonstrate that here on a couple problems below.

00:20 But again, i want to make sure you guys check out this stuff is key.

00:24 We're going to want to have that written down in our notes.

00:27 Okay.

00:27 Okay, so i have in question a, i just have an x1 and an x2.

00:34 So that's telling me i'm going to use the black equation up here.

00:38 And in front of the x1 squared, that is my a1.

00:43 And, ooh, i don't have an x2 squared.

00:47 And so that's going to be a zero, right? i could always write a zero x2 squared here.

00:53 So this is my a two.

00:55 And then finally, the coefficient in front of the x1, x2, that is 2 times a3.

01:03 So 2 times a3 is 5, and so a3 is 5 halves.

01:12 And then to build our matrix, all we do is put the coefficients of the squared terms down the diagonal.

01:20 So that's the 5 and the 0.

01:22 And then the coefficients of the cross terms go in the other spots, 5 halves and 5 halves here.

01:31 Okay, and again what we're doing is we found a way to use matrix multiplication.

01:36 So we're going to say qa of the vector x is x1, x2 as a row vector, times five, five halves, and zero times that matrix.

01:55 And then if i multiply that by x1, x2, that if i do this matrix multiplication, my answer will equal this expression.

02:10 Great.

02:10 Let's look at another one that just has two variables.

02:14 In question b here, i just have an x1 and an x2.

02:19 So again, i'm going to use this top formula here.

02:23 I don't have an x1 squared or an x2 squared.

02:27 So again, i can write those in with a coefficient of zero, and that won't change my expression.

02:36 And so now this zero is my a1, this zero is my a1.

02:40 This zero is my a2.

02:41 And then the cross term here that is two times the a3 term right here.

02:51 So two times a3 is negative 7.

02:54 So a3 is negative 7 halves.

03:00 Build your matrix like it says to do here...

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