Problem 2 - Algebra and Geometry: Introduction to Linear Algebra for Science & Engineering

Question

For each of the following symmetric matrices, find an orthogonal matrix \(P\) and diagonal matrix \(D\) such that \(P^{T} A P=D\). (a) \(A=\left[\begin{array}{rr}1 & -2 \\ -2 & 1\end{array}\right]\) (b) \(A=\left[\begin{array}{rr}5 & 3 \\ 3 & -3\end{array}\right]\) (c) \(A=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right]\) (d) \(A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & -1 & -2 \\ -2 & -2 & 0\end{array}\right]\) (e) \(A=\left[\begin{array}{rrr}1 & 8 & 4 \\ 8 & 1 & -4 \\ 4 & -4 & 7\end{array}\right]\)

Step-by-Step Solution

Answer

For each matrix, find eigenvalues, corresponding eigenvectors, form orthogonal matrix \( P \), then diagonal matrix \( D \). Ensure \( P^T A P = D \).

Step 1: Find Eigenvalues of A

For each matrix, solve the characteristic equation \( \det(A - \lambda I) = 0 \) to find the eigenvalues. Here, \( \lambda \) represents the eigenvalues of the matrix \( A \), and \( I \) is the identity matrix of the same order as \( A \).

Step 2: Find Eigenvectors of A

Once you have the eigenvalues from Step 1, find the corresponding eigenvectors by solving \( (A - \lambda I)x = 0 \) for each eigenvalue \( \lambda \). This involves substituting each eigenvalue into this equation and solving for the vector \( x \).

Step 3: Form Matrix P

Normalize the eigenvectors obtained so that they form an orthonormal set. The matrix \( P \) is formed by using these orthonormal eigenvectors as its columns. An orthogonal matrix \( P \) satisfies \( P^T P = I \), meaning the columns of \( P \) are orthonormal.

Step 4: Form Matrix D

Construct the diagonal matrix \( D \) using the eigenvalues found in Step 1. Place each eigenvalue in a diagonal position corresponding to its eigenvector's position in \( P \). This will ensure that \( P^T A P = D \).

Step 5: Verify the Decomposition

Ensure that the decomposition is correct by performing the computation \( P^T A P \) and checking if the resulting matrix is indeed \( D \). Both \( P^T A P \) and \( D \) should be diagonal matrices with the same diagonal elements as the eigenvalues of \( A \).

Key Concepts

Eigenvectors

Understanding eigenvectors is essential when working with matrices in linear algebra. An eigenvector of a matrix is a special vector that only changes by a scalar factor when that matrix is applied to it. In simpler terms, if you multiply matrix \( A \) by a vector \( x \), and the result is simply the same vector scaled by some number \( \lambda \), then \( x \) is an eigenvector and \( \lambda \) is the corresponding eigenvalue.

To find eigenvectors, once the eigenvalues are determined, we solve the equation \((A - \lambda I)x = 0\), where \( A \) is the matrix, \( I \) is the identity matrix, \( \lambda \) is the eigenvalue, and \( x \) is the eigenvector to be found. This equation ensures that the matrix applied to the vector results in a scaled version of itself.

When solving for eigenvectors:

• Determine the characteristic polynomial and find its roots – these roots are the eigenvalues.
• Use each eigenvalue to find a unique eigenvector by plugging back into the eigenvector equation.
• Check if the vectors are linearly independent for further use in forming orthogonal matrices.

Eigenvectors play a key role when it comes to diagonalizing matrices and understanding transformations in space.

Symmetric Matrices

Symmetric matrices are a special class of matrices with unique properties. A matrix is symmetric if it is equal to its transpose. Simply put, if you flip the matrix across its main diagonal, you get the same matrix. This implies that the elements are mirrored across the diagonal.

The importance of symmetric matrices in linear algebra arises from their nice properties:

• All eigenvalues of symmetric matrices are real numbers.
  
• They always have a set of orthonormal eigenvectors.
• Symmetric matrices can always be diagonalized using an orthogonal matrix.

These characteristics make symmetric matrices highly desirable in applications such as physics, where symmetry often implies conservation laws, and computer graphics, which frequently involve transformation matrices that benefit from these properties.

To check if a matrix is symmetric, simply compare it to its transpose. This is a quick and straightforward process that guarantees the above properties are applicable.

Diagonalization

Diagonalization is an impactful process in linear algebra where a matrix is expressed in terms of a diagonal matrix. A diagonal matrix has all non-zero elements concentrated along its main diagonal and zero everywhere else. The diagonal elements are the eigenvalues of the matrix.

For a matrix to be diagonalizable, it must have the full set of linearly independent eigenvectors equal to its dimension. Here's how to diagonalize a symmetric matrix:

• Find the eigenvalues and eigenvectors of the matrix.
• Use the eigenvectors to form the columns of an orthogonal matrix \( P \).
• The diagonal matrix \( D \) is formed using the eigenvalues on its diagonal.

The relationship between these matrices is expressed as \( P^T A P = D \), meaning the original matrix \( A \) is similar to the diagonal matrix \( D \).

Diagonalization simplifies matrix computations, like calculating powers of matrices or solving differential equations, since working with diagonal matrices is much easier.

Orthogonal Matrix

An orthogonal matrix is a matrix with a special property: its transpose is also its inverse. In simpler terms, when you multiply an orthogonal matrix by its transpose, you get the identity matrix. This property ensures that the columns (and rows) are orthonormal vectors.

Orthogonal matrices emerge naturally in the diagonalization of symmetric matrices. When you have a set of orthonormal eigenvectors, you use them to construct the orthogonal matrix \( P \). This matrix helps transform the original matrix into a diagonal form.

• Orthogonal matrices preserve lengths and angles, hence they are ideal for rotation and reflection transformations.
• They ensure numerical stability in computations.
• Applying an orthogonal matrix to a symmetric matrix maintains its symmetry.

The orthogonal matrix is essential in maintaining the form and properties of data during transformations, making it an invaluable tool in fields such as computer vision, data science, and more.