Question

The quadratic form correspondiny to the symmetric matrix \(\left[\begin{array}{rr}1 & 2 \\ 2 & -4\end{array}\right]\) is .........

Short answer

The quadratic form is \( x_1^2 + 4x_1x_2 - 4x_2^2 \).

Step-by-step solution

Understand the Quadratic Form

The quadratic form associated with a symmetric matrix is given by the expression \( x^T A x \), where \( x \) is a vector, and \( A \) is a symmetric matrix. In this case, the matrix \( A \) is provided as \( \left[\begin{array}{rr}1 & 2 \ 2 & -4\end{array}\right] \).

Define the Vector

Let the vector \( x \) be defined as \( x = \left[ \begin{array}{c} x_1 \ x_2 \end{array} \right] \). This vector represents the variables in the quadratic expression.

Compute the Quadratic Form

The quadratic form \( x^T A x \) is calculated as follows:\[ x^T A x = \left[ \begin{array}{cc} x_1 & x_2 \end{array} \right] \left[ \begin{array}{rr} 1 & 2 \ 2 & -4 \end{array} \right] \left[ \begin{array}{c} x_1 \ x_2 \end{array} \right] \]First, perform the matrix multiplication:1. Multiply \( \left[ \begin{array}{cc} x_1 & x_2 \end{array} \right] \) by the matrix \( \left[ \begin{array}{rr} 1 & 2 \ 2 & -4 \end{array} \right] \): - The result is \( \left[ \begin{array}{cc} 1x_1 + 2x_2 & 2x_1 - 4x_2 \end{array} \right] \).2. Multiply this result with \( \left[ \begin{array}{c} x_1 \ x_2 \end{array} \right] \): - The quadratic form becomes \( (1x_1 + 2x_2)x_1 + (2x_1 - 4x_2)x_2 \).

Simplify the Expression

Simplify the expression obtained in the previous step:\[(1x_1 + 2x_2)x_1 + (2x_1 - 4x_2)x_2 = x_1^2 + 2x_1x_2 + 2x_1x_2 - 4x_2^2\]Combine like terms:\[x_1^2 + 4x_1x_2 - 4x_2^2\].

State the Quadratic Form

The quadratic form of the given symmetric matrix is:\[x_1^2 + 4x_1x_2 - 4x_2^2\].

Key concepts

Symmetric Matrices

Symmetric matrices are a special type of matrix where the elements are mirrored along the main diagonal. This means, for a matrix \( A = [a_{ij}] \), the condition \( a_{ij} = a_{ji} \) holds true. For example, in our case, the matrix \( \left[\begin{array}{rr}1 & 2 \ 2 & -4\end{array}\right] \) is symmetric because the element in the first row, second column (which is 2) is equal to the element in the second row, first column. The advantage of symmetric matrices is that they simplify computations in many problems, like quadratic forms or eigenvalue problems.

Symmetric matrices are particularly useful because they always have real eigenvalues, and their eigenvectors are orthogonal. These properties make them extensively useful in optimization problems and in physical systems like vibrations or electrical circuits.

• Symmetric matrices have properties that simplify processes in linear algebra.
• They make solving certain equations more straightforward due to their structural features.

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra, where you multiply two matrices to produce a third one. The key point is that the number of columns in the first matrix must match the number of rows in the second. In our example, we multiply matrices as part of forming the quadratic form. Here, we need to multiply a vector by a symmetric matrix, and then by another vector.

When multiplying matrices, the entry in the resulting matrix at position \((i, j)\) is found by taking the dot product of the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix. This step-by-step process involves:

• Multiplying corresponding entries and summing them up for each row/column pair.
• The resulting matrix from \(\left[\begin{array}{cc} x_1 & x_2 \end{array}\right] \cdot \left[\begin{array}{rr} 1 & 2 \ 2 & -4 \end{array}\right]\) gives \(\left[\begin{array}{cc} 1x_1 + 2x_2 & 2x_1 - 4x_2 \end{array}\right]\).
• Finally, multiply the resulting vector with \(\left[\begin{array}{c} x_1 \ x_2 \end{array}\right]\) to get the quadratic form.

This systematic method is very useful not only for computations involving vectors and symmetric matrices but also for various tasks in computer graphics, economics, and more.

Vector Representation

Vectors are basic structures in mathematics and physics, viewed as arrows in a space that have both direction and magnitude. Here, we represent variables in our quadratic form as a vector \( x = \left[ \begin{array}{c} x_1 \ x_2 \end{array} \right] \). This representation is vital in expressing many concepts such as displacements, forces, or even more abstract things as sets of numbers.

In the context of quadratic forms, using vectors is significant because:

• They simplify the representation of multiple variables in a concise way.
• They facilitate operations such as dot products, matrix multiplications, and linear transformations.
• In this problem, they allow us to neatly express and calculate expressions like \( x^T A x \).

The use of vectors not only makes the computation more systematic but also ties into more extensive applications such as computer simulations, physics equations, and real-world problem-solving scenarios.