Question

If \(\mathbf{x}=\left(x_{1}, \ldots, x_{n}\right)^{T}\) is a column of variables, \(A=A^{T}\) is \(n \times n, B\) is \(1 \times n,\) and \(c\) is a constant, \(\mathbf{x}^{T} A \mathbf{x}+B \mathbf{x}=c\) is called a quadratic equation in the variables \(x_{i}\) a. Show that new variables \(y_{1}, \ldots, y_{n}\) can be found such that the equation takes the form $$\lambda_{1} y_{1}^{2}+\cdots+\lambda_{r} y_{r}^{2}+k_{1} y_{1}+\cdots+k_{n} y_{n}=c$$ b. Put \(x_{1}^{2}+3 x_{2}^{2}+3 x_{3}^{2}+4 x_{1} x_{2}-4 x_{1} x_{3}+5 x_{1}-6 x_{3}=7\) in this form and find variables \(y_{1}, y_{2}, y_{3}\) as in (a).

Short answer

Diagonalize the matrix to find new variables \( y_i \) such that the equation has the desired form.

Step-by-step solution

Express the Quadratic Form

The given quadratic form is \( \mathbf{x}^T A \mathbf{x} + B \mathbf{x} = c \). Rewrite it in a way that separates the quadratic and linear terms: \( Q(\mathbf{x}) + L(\mathbf{x}) = c \), where \( Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} \) is the quadratic form, and \( L(\mathbf{x}) = B \mathbf{x} \) is the linear form.

Diagonalize the Quadratic Form

To find new variables \( y_1, y_2, \ldots, y_n \), diagonalize the matrix \( A \). This can be done by finding an orthogonal matrix \( P \) such that \( P^T A P = D \), where \( D \) is a diagonal matrix with eigenvalues of \( A \) on its diagonal. The transformation \( \mathbf{y} = P^T \mathbf{x} \) will simplify \( Q(\mathbf{x}) \) to \( \lambda_1 y_1^2 + \lambda_2 y_2^2 + \cdots + \lambda_n y_n^2 \).

Transform the Linear Form

Transform the linear term \( L(\mathbf{x}) = B \mathbf{x} \) using the change of variables \( \mathbf{y} = P^T \mathbf{x} \). This gives \( L(\mathbf{x}) = B P \mathbf{y} \). Represent this as a linear combination \( k_1 y_1 + k_2 y_2 + \cdots + k_n y_n \).

Establish Quadratic Equation in New Variables

Combine the transformed quadratic and linear forms in terms of \( y_i \): \( \lambda_1 y_1^2 + \lambda_2 y_2^2 + \ldots + \lambda_r y_r^2 + k_1 y_1 + k_2 y_2 + \ldots + k_n y_n = c \). This shows that \( y_i \) satisfy the conditions given in part (a) of the exercise.

Apply to Given Quadratic Equation

For part (b), determine the matrices and constants in the equation \( x_1^2 + 3x_2^2 + 3x_3^2 + 4x_1x_2 - 4x_1x_3 + 5x_1 - 6x_3 = 7 \). Identify \( A \), the symmetric matrix, and \( B \), the coefficient vector. Here, \( A = \begin{pmatrix} 1 & 2 & -2 \ 2 & 3 & 0 \ -2 & 0 & 3 \end{pmatrix} \) and \( B = (5, 0, -6) \).

Perform Diagonalization and Solve for \( y \, \) Variables

Diagonalize \( A \) to find an orthogonal matrix \( P \) such that \( P^T A P = D \), where \( D \) is diagonal. Transform the original quadratic form using \( \mathbf{y} = P^T \mathbf{x} \). Identify linear terms in new variables \( y_i \): \( B P \mathbf{y} \). Solve for the new variables to express the equation in the desired form. The diagonalization will give the simplified equation with terms in \( \lambda_i y_i^2 \).

Key concepts

Quadratic Form

A quadratic form is an expression involving a sum of squares with coefficients for each variable. It's a way to represent and analyze equations where multiple variables are involved, especially with quadratic terms. In mathematical terms, a quadratic form can be written as \( Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} \), where \( A \) is a symmetric matrix of coefficients, and \( \mathbf{x} \) is a vector of variables. This form groups terms involving squares of the variables and their cross-products.

When analyzing quadratic equations, it's essential to distinguish between the quadratic and linear parts of the expression. Here, the quadratic part is represented by \( \mathbf{x}^T A \mathbf{x} \), while any remaining linear terms are gathered into \( B \mathbf{x} \), together forming the full equation \( \mathbf{x}^T A \mathbf{x} + B \mathbf{x} = c \). Simplifying these forms can make the equation easier to work with, especially when transformations like diagonalization are used.

Diagonalization

Diagonalization is a powerful method for simplifying matrices, specifically symmetric matrices like the ones found in quadratic forms. The goal is to find an orthogonal matrix \( P \), which allows us to convert the original matrix \( A \) into a diagonal matrix \( D \) by transforming it with \( P^T A P = D \). This step simplifies the analysis because the diagonal matrix \( D \) has eigenvalues as its diagonal elements, while off-diagonal elements are zero.

When performing diagonalization, you essentially rotate the original coordinate system using the orthogonal matrix \( P \). This makes it easier to work with multiple-variable expressions by reducing complex coupled terms to simple individual squares. The transformation involves the substitution \( \mathbf{y} = P^T \mathbf{x} \), where \( \mathbf{y} \) are the new variables. This leads to a more manageable form where each quadratic term can be expressed as \( \lambda_i y_i^2 \), with \( \lambda_i \) representing the eigenvalues.

• Diagonalization is particularly useful in revealing the essence of quadratic forms.
• It helps identify fundamental directions, represented by eigenvectors, that simplify the expression.

Eigenvalues

Eigenvalues play a crucial role in the process of diagonalization and the study of quadratic forms. They are the values placed on the diagonal of the matrix \( D \) after diagonalizing \( A \). Eigenvalues, denoted by \( \lambda_i \), provide a measure of the scaling factor along the respective eigenvectors of the matrix.

Finding the eigenvalues involves solving the characteristic equation \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix. Each eigenvalue represents how much each axis of the original space is stretched or compressed during the transformation. This allows quadratic forms to be expressed in the simplified form \( \lambda_1 y_1^2 + \lambda_2 y_2^2 + \cdots + \lambda_n y_n^2 \), making it easier to understand the effects of the quadratic equation on the system of variables.

• Eigenvalues can show stability and dynamics in systems described by quadratic forms.
• They make it possible to determine if the quadratic form is strictly positive, negative, or indefinite.