Problem 15
Question
Determine a set of principal axes for the given quadratic form, and reduce the quadratic form to a sum of squares. $$\mathbf{x}^{T} A \mathbf{x}, \quad A=\left[\begin{array}{ll} 5 & 2 \\ 2 & 5 \end{array}\right]$$
Step-by-Step Solution
Verified Answer
The given quadratic form can be transformed into a sum of squares by first finding the eigenvalues \(\lambda_1 = 3\) and \(\lambda_2 = 7\), and the eigenvectors \(v_1 = \begin{bmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{bmatrix}\) and \(v_2 = \begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix}\). Then we create the matrix P: \[P = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{bmatrix}\] and transform the quadratic form into \(\mathbf{y}^T \mathbf{D} \mathbf{y}\), where \[\mathbf{D} = \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}\]. The resulting sum of squares is \(3y_1^2 + 7y_2^2\).
1Step 1: Compute Eigenvalues and Eigenvectors of Matrix A
To begin, we must find the eigenvalues (\(λ_1\) and \(λ_2\)) and eigenvectors (\(v_1\) and \(v_2\)) of the given matrix A. This can be done by solving the following equation:
\[|\mathbf{A} - λ \mathbf{I}| = 0\]
Where \(A\) is given and \(\mathbf{I}\) is the identity matrix. Plugging the values, we get:
\[|\begin{bmatrix} 5-λ & 2 \\ 2 & 5-λ \end{bmatrix}| = 0\]
On solving the above equation, we find the eigenvalues:
\(λ_1 = 3\) and \(λ_2 = 7\)
Now, we need to compute the eigenvectors associated with these eigenvalues. For each eigenvalue, solve the equation \( (\mathbf{A} - λ \mathbf{I})v = 0 \).
For \(λ_1\):
\[
\begin{bmatrix}
2 & 2 \\
2 & 2
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
= 0\]
On solving, we get the eigenvector \(v_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}\).
For \(λ_2\):
\[
\begin{bmatrix}
-2 & 2 \\
2 & -2
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
= 0\]
On solving, we get the eigenvector \(v_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\).
2Step 2: Normalize Eigenvectors
Next, we need to normalize the eigenvectors to create an orthogonal basis. The normalized eigenvectors can be computed as follows:
\(v_1 = \frac{1}{\sqrt {1^2 + (-1)^2}} \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{bmatrix} \)
\(v_2 = \frac{1}{\sqrt {1^2 + 1^2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix}\)
These normalized eigenvectors now form an orthogonal basis.
3Step 3: Compute the Matrix P
To perform the transformation, form the matrix P with the normalized eigenvectors as its columns:
\[P = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{bmatrix}\]
4Step 4: Transform Quadratic Form to a Sum of Squares
Now we can transform the quadratic form and rewrite it as a sum of squares. The formula to transform is as follows:
\(\mathbf{y}^T \mathbf{D} \mathbf{y} = \mathbf{x}^T \mathbf{A} \mathbf{x}\),
where \(\mathbf{D}\) is the diagonal matrix formed by the eigenvalues, \(\mathbf{y} = \mathbf{P}^T\mathbf{x}\) and \(\mathbf{A}\) is the given matrix.
\[\mathbf{D} = \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}\]
Substituting \(\mathbf{D}\) and \(\mathbf{P}\) into the transformation equation we get:
\[
\begin{bmatrix} y_1 \\ y_2 \end{bmatrix}^T
\begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix}
\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} =
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}^T
\begin{bmatrix} 5 & 2 \\ 2 & 5 \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
\]
This leads to the sum of squares form of the quadratic form:
\(\mathbf{y}^T \mathbf{D} \mathbf{y} = 3y_1^2 + 7y_2^2\)
Thus, we have successfully transformed the given quadratic form into a sum of squares.
Key Concepts
Quadratic FormEigenvalues and EigenvectorsOrthogonal BasisMatrix Transformation
Quadratic Form
A quadratic form is an expression involving a square of variables, commonly represented as \( \mathbf{x}^T A \mathbf{x} \), where \( \mathbf{x} \) is a vector and \( A \) is a square matrix. Understanding quadratic forms is crucial because they appear frequently in various fields, including optimization and statistics. For a matrix \( A \), the quadratic form combines the interactions between variables, typically resulting in an indication of the direction and curvature of a function.
In the specific exercise, the quadratic form is provided as \( \mathbf{x}^T A \mathbf{x} \) with \( A = \begin{bmatrix} 5 & 2 \ 2 & 5 \end{bmatrix} \). Identifying principal axes will simplify this form into a more understandable mathematical expression. Analyzing how matrices and vectors contribute to the quadratic form allows for deeper insight into mathematical modeling and prediction.
In the specific exercise, the quadratic form is provided as \( \mathbf{x}^T A \mathbf{x} \) with \( A = \begin{bmatrix} 5 & 2 \ 2 & 5 \end{bmatrix} \). Identifying principal axes will simplify this form into a more understandable mathematical expression. Analyzing how matrices and vectors contribute to the quadratic form allows for deeper insight into mathematical modeling and prediction.
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are important mathematical concepts that help in understanding the transformations associated with a quadratic form. An eigenvalue is a scalar that indicates the magnitude of an eigenvector’s transformation, while an eigenvector indicates the direction. To find them, we solve the characteristic equation \(|\mathbf{A} - \lambda \mathbf{I}| = 0\), where \( \lambda \) are the eigenvalues and \( \mathbf{I} \) is the identity matrix.
For the matrix \( A \) in the original problem, solving the characteristic equation yields eigenvalues \( \lambda_1 = 3 \) and \( \lambda_2 = 7 \). Once eigenvalues are known, we calculate the eigenvectors by solving the equation \( (\mathbf{A} - \lambda \mathbf{I})v = 0 \) for each eigenvalue.
For the matrix \( A \) in the original problem, solving the characteristic equation yields eigenvalues \( \lambda_1 = 3 \) and \( \lambda_2 = 7 \). Once eigenvalues are known, we calculate the eigenvectors by solving the equation \( (\mathbf{A} - \lambda \mathbf{I})v = 0 \) for each eigenvalue.
- For \( \lambda_1 \): Eigenvector \( v_1 = \begin{bmatrix} 1 \ -1 \end{bmatrix} \)
- For \( \lambda_2 \): Eigenvector \( v_2 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \)
Orthogonal Basis
Creating an orthogonal basis can greatly simplify quadratic forms and matrix computations by reducing complexity. An orthogonal basis comprises vectors that are perpendicular to each other—this means the dot product between any pair of different vectors is zero. In contexts involving quadratic forms, having orthogonal vectors aids in simplifying the problems, such as transforming a matrix to its diagonal form.
To create an orthogonal basis from the given eigenvectors, we normalize each one. Normalization is the process of scaling a vector to have a unit length by dividing it by its norm:
To create an orthogonal basis from the given eigenvectors, we normalize each one. Normalization is the process of scaling a vector to have a unit length by dividing it by its norm:
- Eigenvector \( v_1 = \begin{bmatrix} 1/\sqrt{2} \ -1/\sqrt{2} \end{bmatrix} \)
- Eigenvector \( v_2 = \begin{bmatrix} 1/\sqrt{2} \ 1/\sqrt{2} \end{bmatrix} \)
Matrix Transformation
Matrix transformation is a powerful mathematical tool used to change the basis of a quadratic form. By transforming the quadratic form, it's possible to rewrite it in terms of a different set of variables, often making it easier to work with. In this exercise, matrix transformation involves using the matrix \( P \), composed of normalized eigenvectors, to alter the original quadratic form.
The transformation equation \( \mathbf{y}^T \mathbf{D} \mathbf{y} = \mathbf{x}^T \mathbf{A} \mathbf{x} \) illustrates how the form is rewritten. Here, \( \mathbf{D} \) is a diagonal matrix composed of our eigenvalues, and \( \mathbf{y} = \mathbf{P}^T\mathbf{x} \).
This process yields a sum of squares form \( 3y_1^2 + 7y_2^2 \), simplifying the expression and focusing on its key components. Matrix transformations, when used thoughtfully, enhance the understanding and efficiency of dealing with complex mathematical models.
The transformation equation \( \mathbf{y}^T \mathbf{D} \mathbf{y} = \mathbf{x}^T \mathbf{A} \mathbf{x} \) illustrates how the form is rewritten. Here, \( \mathbf{D} \) is a diagonal matrix composed of our eigenvalues, and \( \mathbf{y} = \mathbf{P}^T\mathbf{x} \).
This process yields a sum of squares form \( 3y_1^2 + 7y_2^2 \), simplifying the expression and focusing on its key components. Matrix transformations, when used thoughtfully, enhance the understanding and efficiency of dealing with complex mathematical models.
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