Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 20
Use the routine MATCOMP in LINTEK, or MATLAB, t\cap find a diagonal form into which the given form can be transformed by an orthogonal substitution. Do not give the substitution. x^{2}-8 x y+y^{2}
Short Answer
Expert verified
Diagonal form: \( \begin{pmatrix} -3 & 0 \\ 0 & 5 \end{pmatrix} \).
Step by step solution
01
Understand the Problem
We are given a quadratic form \( x^2 - 8xy + y^2 \) and need to find its diagonal form using an orthogonal substitution. The goal is to transform the matrix representation of the quadratic form into a diagonal matrix using an orthogonal matrix.
02
Express the Quadratic Form as a Matrix
The given quadratic form \( x^2 - 8xy + y^2 \) can be represented in matrix form as \( \mathbf{x}^T A \mathbf{x} \), where \( A \) is a symmetric matrix. For this quadratic, the matrix \( A \) can be written as: \[ A = \begin{pmatrix} 1 & -4 \ -4 & 1 \end{pmatrix} \]
03
Perform Eigenvalue Decomposition
To diagonalize the matrix \( A \) using an orthogonal transformation, find the eigenvalues and eigenvectors of \( A \). Calculate the determinant of \( A - \lambda I \) and solve the characteristic equation to find eigenvalues \( \lambda_1 \) and \( \lambda_2 \).
04
Find Eigenvalues
Calculate the eigenvalues from the characteristic equation \( \text{det}(A - \lambda I) = 0 \). For our matrix \( A \), the characteristic determinant is:\[\begin{vmatrix} 1 - \lambda & -4 \ -4 & 1 - \lambda \end{vmatrix} = (1 - \lambda)^2 - (-4)^2 = \lambda^2 - 2\lambda - 15 = 0\] Solving, we find the eigenvalues to be \( \lambda_1 = -3 \) and \( \lambda_2 = 5 \).
05
Find Eigenvectors
For each eigenvalue, determine the corresponding eigenvector by solving \( (A - \lambda I) \mathbf{v} = 0 \). For \( \lambda_1 = -3 \), solve:\[ (\begin{pmatrix} 4 & -4 \ -4 & 4 \end{pmatrix}) \mathbf{v} = 0 \]A possible eigenvector is \( \mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix} \). For \( \lambda_2 = 5 \), solve:\[ (\begin{pmatrix} -4 & -4 \ -4 & -4 \end{pmatrix}) \mathbf{v} = 0 \]A possible eigenvector is \( \mathbf{v}_2 = \begin{pmatrix} 1 \ -1 \end{pmatrix} \).
06
Form Orthogonal Matrix P
Normalize the eigenvectors, if necessary, and construct the orthogonal matrix \( P \) using these eigenvectors as columns, forming \( P = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix} \). Ensure that \( P^T P = I \) to verify orthogonality.
07
Diagonalize the Matrix
The diagonal form of the matrix \( A \) is given by \( D = P^T A P \). Calculate \( D \) to verify that it is a diagonal matrix with the eigenvalues on the diagonal: \[ D = \begin{pmatrix} -3 & 0 \ 0 & 5 \end{pmatrix} \].
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Orthogonal Substitution
Orthogonal substitution is a powerful mathematical technique used to simplify quadratic forms. When given a quadratic form like \( x^2 - 8xy + y^2 \), you can transform it into a simpler form using orthogonal matrices. An orthogonal matrix is characterized by the property \( P^T P = I \), where \( P^T \) is the transpose of \( P \) and \( I \) is the identity matrix. This transformation helps in "rotating" the axes of the quadratic form without altering its inherent geometry.
- Orthogonal matrices have orthonormal vectors as their columns.
- They maintain distances and angles due to their orthogonality property.
- Using an orthogonal substitution, complex cross-terms like \(-8xy\) can be eliminated, simplifying the expression.
Eigenvalue Decomposition
Eigenvalue decomposition is a technique where a matrix is expressed in terms of its eigenvalues and eigenvectors. For our quadratic form's matrix representation, this involves finding the eigenvalues and constructing a diagonal matrix from them. Given the matrix \( A = \begin{pmatrix} 1 & -4 \ -4 & 1 \end{pmatrix} \), performing eigenvalue decomposition allows us to express \( A \) as \( PDP^{-1} \), where \( D \) is a diagonal matrix.
- Eigenvalues, \( \lambda_1 \) and \( \lambda_2 \), represent the roots of the characteristic equation.
- The matrix \( D \) is constructed using these eigenvalues, listing them on its diagonal.
- Eigenvalue decomposition is fundamental for understanding how linear transformations scale vectors.
Quadratic Form
A quadratic form is an expression involving a symmetrical square matrix that represents a quadratic polynomial. In two variables, it generally takes the form \( ax^2 + 2bxy + cy^2 \). However, it can be expressed in matrix notation as \( \mathbf{x}^T A \mathbf{x} \), where \( \mathbf{x} \) is a vector of variables \( (x, y) \) and \( A \) is a symmetric matrix.
- The symmetry of the matrix ensures that the form can be manipulated using linear algebra techniques.
- Diagonalizing the quadratic form helps in understanding its geometric representation.
- It finds applications in optimization and critical point analysis.
Eigenvectors
Eigenvectors are non-zero vectors that only scale when a linear transformation is applied, without changing their direction. In terms of matrices, when you have a matrix \( A \) and an eigenvalue \( \lambda \), any vector \( \mathbf{v} \) that satisfies \( A\mathbf{v} = \lambda\mathbf{v} \) is called an eigenvector of \( A \).
- Each eigenvector corresponds to a specific eigenvalue.
- They provide a fundamental insight into the matrix's behavior during transformations.
- In the quadratic form diagonalization, eigenvectors define the orientation of the transformed coordinate system.
