Problem 43 - Advanced Engineering Mathematics

Question

Verify that the quadratic form \(a x^{2}+b x y+c y^{2}\) is the same as $$ \left(\begin{array}{ll} x & y \end{array}\left(\begin{array}{rr} a & \frac{1}{2} b \\ \frac{1}{2} b & c \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right) .\right. $$

Step-by-Step Solution

Answer

The quadratic form is equivalent to the given matrix product.

Step 1: Understand the Exercise

The exercise asks us to verify that a given quadratic expression \(a x^{2}+b x y+c y^{2}\) can be expressed as a matrix product involving vectors and a symmetric matrix. Our task is to break this down and check the equivalence of the matrix formulation with the quadratic form.

Step 2: Set Up the Quadratic Form

The quadratic form is \(a x^{2}+b x y+c y^{2}\). This expression involves coefficients \(a\), \(b\), and \(c\) for the terms \(x^2\), \(xy\), and \(y^2\), respectively.

Step 3: Identify the Matrix Representation

We need to verify the expression using the matrix representation \[\begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} a & \frac{1}{2} b \ \frac{1}{2} b & c \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}\] where the matrix is symmetric with elements \(a\), \(\frac{1}{2} b\), and \(c\).

Step 4: Perform Matrix Multiplication

Start by multiplying the matrix by the vector on the right: \[ \begin{pmatrix} a & \frac{1}{2} b \ \frac{1}{2} b & c \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} ax + \frac{1}{2} by \ \frac{1}{2} bx + cy \end{pmatrix} \]

Step 5: Compute Dot Product

Now compute the dot product with the vector on the left: \[ \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} ax + \frac{1}{2} by \ \frac{1}{2} bx + cy \end{pmatrix} = x(ax + \frac{1}{2} by) + y(\frac{1}{2} bx + cy) \] This results in \( ax^2 + \frac{1}{2} bxy + \frac{1}{2} bxy + cy^2 \).

Step 6: Simplify and Verify

Simplify the expression by combining like terms:\( ax^2 + bxy + cy^2 \). This is exactly the original quadratic form \( a x^{2}+b x y+c y^{2} \). Hence, the matrix representation is verified.

Key Concepts

Matrix Representation

The idea of a matrix representation is critical when converting a quadratic form into a format that involves matrices and vectors. Imagine taking a quadratic expression like \( a x^{2} + b x y + c y^{2} \) and expressing it as a matrix operation. To achieve this, identify a vector

• \( \begin{pmatrix} x \ y \end{pmatrix} \)

and a symmetric matrix

• \( \begin{pmatrix} a & \frac{1}{2}b \ \frac{1}{2}b & c \end{pmatrix} \).

To represent the quadratic form using matrices, simply arrange the vectors and matrices in the following way:

• \( \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} a & \frac{1}{2} b \ \frac{1}{2} b & c \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \).

This setup transforms the quadratic equation into a format that can be tackled using matrix mathematics.

Symmetric Matrix

A symmetric matrix is a type of matrix that is identical when mirrored along its main diagonal. This means the elements above the diagonal are the same as those below. For instance, the matrix

• \( \begin{pmatrix} a & \frac{1}{2} b \ \frac{1}{2} b & c \end{pmatrix} \)

is symmetric because the upper right element \( \frac{1}{2} b \) is equal to the lower left one. Symmetric matrices commonly arise in quadratic forms and have properties that simplify calculations. They ensure that the system behaves predictably under various mathematical operations, making it easier to understand and analyze the relationships between variables.

Dot Product

The dot product is a term used to describe an operation between two vectors. In the context of quadratic forms, it plays a pivotal role in simplifying expressions. When you have vectors like

• \( \begin{pmatrix} x & y \end{pmatrix} \)

and

• \( \begin{pmatrix} ax + \frac{1}{2} by \ \frac{1}{2} bx + cy \end{pmatrix} \),

you can perform a dot product by multiplying corresponding elements and summing the results:

• \( x(ax + \frac{1}{2} by) + y(\frac{1}{2} bx + cy) \).

This process helps in evaluating and simplifying expressions, ultimately leading us back to the original quadratic form \( ax^2 + bxy + cy^2 \). It reveals how the geometry of vector multiplication relates to the structure of quadratic equations.

Matrix Multiplication

Matrix multiplication is a critical concept in translating quadratic forms into matrix language. It involves multiplying matrices with vectors to transform and simplify expressions. In our case, start by multiplying a symmetric matrix

• \( \begin{pmatrix} a & \frac{1}{2} b \ \frac{1}{2} b & c \end{pmatrix} \)

by a column vector

• \( \begin{pmatrix} x \ y \end{pmatrix} \),

producing another column vector:

• \( \begin{pmatrix} ax + \frac{1}{2} by \ \frac{1}{2} bx + cy \end{pmatrix} \).

Next, input this vector into a dot product with a row vector

• \( \begin{pmatrix} x & y \end{pmatrix} \).

This structured series of operations leverages matrix properties to translate more complex expressions, ultimately verifying the equivalence of formats in a straightforward and efficient manner.