Problem 1 - Algebra and Geometry: Introduction to Linear Algebra for Science & Engineering

Question

Determine the quadratic form corresponding to the given symmetric matrix. (a) \(A=\left[\begin{array}{rr}1 & 3 \\ 3 & -1\end{array}\right]\) (b) \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & -2 & 3 \\ 0 & 3 & -1\end{array}\right]\) \text { (c) } A=\left[\begin{array}{rrr} -2 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & 0 \end{array}\right]

Step-by-Step Solution

Answer

(a) \(Q(x) = x_1^2 + 6x_1x_2 - x_2^2\); (b) \(Q(x) = x_1^2 - 2x_2^2 + 6x_2x_3 - x_3^2\); (c) \(Q(x) = -2x_1^2 + 2x_1x_2 + 2x_1x_3 + x_2^2 - 2x_2x_3\).

Step 1: Understanding the Matrix and Variables

For any symmetric matrix \( A \) of size \( n \times n \), the quadratic form is expressed as \( Q(x) = x^T A x \), where \( x \) is a vector with components \( x_1, x_2, \dots, x_n \). For example, for a 2x2 matrix, \( x = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) and for a 3x3 matrix, \( x = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \).

Step 1: Quadratic Form for Matrix (a)

Given the matrix \[ A = \begin{bmatrix} 1 & 3 \ 3 & -1 \end{bmatrix} \]we calculate the quadratic form using \[ Q(x) = \begin{bmatrix} x_1 & x_2 \end{bmatrix} \begin{bmatrix} 1 & 3 \ 3 & -1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \].Calculate the expression: \[ Q(x) = x_1^2 + 3x_1x_2 + 3x_2x_1 - x_2^2 = x_1^2 + 6x_1x_2 - x_2^2 \].

Step 2: Quadratic Form for Matrix (b)

Given the matrix\[ A = \begin{bmatrix} 1 & 0 & 0 \ 0 & -2 & 3 \ 0 & 3 & -1 \end{bmatrix} \]we calculate the quadratic form using\[ Q(x) = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \ 0 & -2 & 3 \ 0 & 3 & -1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \].Calculate the expression:\[ Q(x) = x_1^2 - 2x_2^2 + 3x_2x_3 + 3x_3x_2 - x_3^2 = x_1^2 - 2x_2^2 + 6x_2x_3 - x_3^2 \].

Step 3: Quadratic Form for Matrix (c)

Given the matrix\[ A = \begin{bmatrix} -2 & 1 & 1 \ 1 & 1 & -1 \ 1 & -1 & 0 \end{bmatrix} \]we calculate the quadratic form using\[ Q(x) = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} \begin{bmatrix} -2 & 1 & 1 \ 1 & 1 & -1 \ 1 & -1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \].Calculate the expression:\[ Q(x) = -2x_1^2 + 2x_1x_2 + 2x_1x_3 + x_2^2 - 2x_2x_3 \].

Key Concepts

Symmetric Matrix

A symmetric matrix is a type of square matrix that remains unchanged when its rows are swapped with columns. This means a matrix is symmetric if it equals its own transpose. For example, for any element in a matrix located at row i, column j, denoted as \( a_{ij} \), it should be equal to the element at row j, column i, \( a_{ji} \).
To put it simply, if you mirror the matrix along its main diagonal (from the top left to the bottom right), you will see the same values on both sides.

• The main diagonal elements (from top left to bottom right) may or may not be zero.
• All symmetric matrices are square by definition, which means they must have the same number of rows and columns.

Symmetric matrices are significant, especially in physics and computer science, because they often represent real-world phenomena like geometry and network flows. In the field of linear algebra, symmetric matrices also help in simplifying computations related to quadratic forms, as they guarantee real eigenvalues.

Matrix Multiplication

Matrix multiplication is a method of combining two matrices to produce a third matrix. It is not as straightforward as multiplying numbers; the order and size of matrices must be compatible. You can only multiply two matrices if the number of columns in the first matrix equals the number of rows in the second matrix.
Here's how it works: to find an element in the resulting product matrix at position \( (i,j) \), multiply each element of the i-th row of the first matrix with the corresponding j-th column element of the second matrix and sum them up. So, for matrices \( A \) and \( B \), the element \( c_{ij} \) of the resulting matrix \( C \) would be: \[ c_{ij} = ext{sum of } (A_{ik} imes B_{kj}) \text{ for all } k. \]

• Matrix multiplication is not commutative: \( AB \) might not equal \( BA \).
• Matrix multiplication is associative: \((AB)C = A(BC)\).
• It's also distributive: \( A(B+C) = AB + AC \).

This operation is pivotal in solving systems of linear equations, transformations, and operations involving symmetric matrices, as it is used to compute properties like the quadratic form.

Linear Algebra

Linear Algebra is a branch of mathematics that studies vectors, vector spaces (linear spaces), linear transformations, and systems of linear equations. It is foundational for many areas including science, engineering, computer science, and more.
At its core, Linear Algebra uses matrices and vectors to represent and solve systems of linear equations, allowing for the understanding of geometric and numerical relationships. Here are the key components:

• **Vectors**: Objects that have both a magnitude and direction. Can be represented as column matrices.
• **Matrix**: A rectangular array of numbers arranged in rows and columns. Used for linear transformations.
• **Determinants**: A value calculated from a square matrix that indicates if it is invertible.
• **Eigenvalues and Eigenvectors**: Used to understand linear transformations, especially in symmetric matrices.

Linear Algebra isn't just limited to theoretical applications but is extensively used in computer graphics, machine learning, data science, and optimization problems due to its ability to represent and solve complex real-world problems. Concepts like quadratic forms often emerge in these applications, showcasing the utility of Linear Algebra in expressing and solving multi-variable systems.