Question

For each matrix \(A,\) find an orthogonal matrix \(P\) such that \(P^{-1} A P\) is diagonal. a. \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) b. \(A=\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right]\) c. \(A=\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 2 & 5\end{array}\right]\) d. \(A=\left[\begin{array}{lll}3 & 0 & 7 \\ 0 & 5 & 0 \\ 7 & 0 & 3\end{array}\right]\) e. \(A=\left[\begin{array}{lll}1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right]\) f. \(A=\left[\begin{array}{rrr}5 & -2 & -4 \\ -2 & 8 & -2 \\ -4 & -2 & 5\end{array}\right]\) g. \(A=\left[\begin{array}{llll}5 & 3 & 0 & 0 \\ 3 & 5 & 0 & 0 \\ 0 & 0 & 7 & 1 \\ 0 & 0 & 1 & 7\end{array}\right]\) h. \(A=\left[\begin{array}{rrrr}3 & 5 & -1 & 1 \\ 5 & 3 & 1 & -1 \\ -1 & 1 & 3 & 5 \\ 1 & -1 & 5 & 3\end{array}\right]\)

Short answer

Find matrix \( P \) from eigenvectors; \( P^{-1} A P \) will be diagonal.

Step-by-step solution

Understand the Objective

The goal is to find an orthogonal matrix \( P \) such that \( P^{-1} A P \) is diagonal. This means that \( P \) should consist of the eigenvectors of \( A \), and the diagonal elements of the diagonal matrix will be the eigenvalues of \( A \).

Eigenvalues and Eigenvectors Calculation (General)

To find \( P \), we first need to compute the eigenvalues and eigenvectors of each matrix \( A \). Given that \( A \) is a real symmetric matrix, it will have real eigenvalues and orthogonal eigenvectors, which can be arranged as columns of the orthogonal matrix \( P \).

Solve for Matrix (a)

Find the eigenvalues of matrix \( A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \) by solving the characteristic equation \( \det(\lambda I - A) = 0 \). The eigenvalues are \( \lambda = 1 \) and \( \lambda = -1 \). The corresponding eigenvectors are \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \) and \( \begin{bmatrix} 1 \ -1 \end{bmatrix} \), respectively. Normalize these vectors to form the orthogonal matrix \( P \).

Solve for Matrix (b)

For \( A = \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} \), the eigenvalues found by solving \( \det(\lambda I - A) = 0 \) are \( \lambda = 0 \) and \( \lambda = 2 \). The corresponding normalized eigenvectors give us the orthogonal matrix \( P \).

Solve for Matrices (c) to (h)

Repeat the process for matrices (c) through (h): - Find the eigenvalues by solving the characteristic polynomial \( \det(\lambda I - A) = 0 \).- Determine the corresponding eigenvectors.- Construct an orthonormal set of eigenvectors, which will form the columns of matrix \( P \).- Normalize these vectors to ensure that the matrix \( P \) is orthogonal.

Construct Orthogonal Matrix \( P \)

For each subproblem, construct \( P \) from the normalized eigenvectors. Verify the orthogonality by ensuring that \( P^TP = I \), where \( I \) is the identity matrix.

Verify Solution

Check that \( P^{-1} A P \) results in a diagonal matrix with the diagonal consisting of eigenvalues. Since \( P \) is orthogonal, \( P^{-1} = P^T \), thus \( P^T A P \) should be computed.

Key concepts

Eigenvalues

Eigenvalues are fundamental in understanding the properties of matrices. When working with a matrix, we often want to see how it can be broken down into simpler parts. This is where eigenvalues come in. They are special numbers associated with a matrix that provide insight into the matrix's behavior, such as its stability and variance.
To find eigenvalues, we solve the characteristic equation, which is derived from the determinant of the matrix subtracted by a scalar times the identity matrix: \( \det(\lambda I - A) = 0 \). This equation can have \

• Real solutions.
• Complex solutions.

When a matrix is symmetric, all eigenvalues will be real, simplifying the process of diagonalization. Each eigenvalue tells us how much the corresponding eigenvector is stretched or compressed along its direction. These values are the diagonal elements of the diagonal matrix achieved in diagonalization. By identifying eigenvalues, we take the first step towards transforming a complex matrix into a simple and understandable form.

Eigenvectors

Eigenvectors complement eigenvalues in the process of diagonalizing a matrix. For a given eigenvalue \( \lambda \), an eigenvector \( \mathbf{v} \) is a non-zero vector that satisfies the equation \( A\mathbf{v} = \lambda\mathbf{v} \). This means that the vector is only scaled by the action of the matrix, not rotated or otherwise altered.
By finding the eigenvectors of a matrix, we can form the orthogonal matrix \( P \), whose columns are these eigenvectors. This matrix \( P \) helps in transforming the original matrix \( A \) into its diagonal form. The eigenvectors are crucial because:

• They form the basis for the vector space.
• The directions along which the matrix acts by simply stretching or compressing.

Eigenvectors are not unique, as any multiple of an eigenvector is also an eigenvector due to the nature of their definition. In symmetric matrices, which we are dealing with, eigenvectors associated with different eigenvalues are orthogonal to each other. This orthogonality plays a key role in the creation of the orthogonal matrix \( P \). Once normalized, these eigenvectors ensure that \( P \) is orthonormal, a property essential for diagonalization.

Orthogonal Matrix

An orthogonal matrix is a special type of square matrix characterized by the property that its transpose is equal to its inverse. In simple terms, if \( P \) is an orthogonal matrix, then \( P^T P = I \), where \( I \) is the identity matrix. This property ensures that transformations performed using orthogonal matrices preserve angles and lengths.
Orthogonal matrices are used in diagonalization to ensure that \( P^{-1} = P^T \), which simplifies calculations such as \( P^{-1} A P = P^T A P \). The fact that lengths and angles remain unchanged during transformations makes these matrices extremely useful in practical applications.
The orthogonal matrix \( P \) is constructed using normalized eigenvectors of the given matrix \( A \). Normalization makes these vectors unit vectors, thus contributing to the orthonormality of matrix \( P \). As a result, orthogonal matrices provide:

• Ease of computation.
• Geometric interpretation.
• Stability in numerical calculations.

Symmetric Matrices

Symmetric matrices are crucial in the context of diagonalization because they exhibit properties that facilitate easy computation of eigenvalues and eigenvectors. A symmetric matrix \( A \) is one where \( A = A^T \). This structure ensures that the matrix is equal to its transpose and leads to some valuable outcomes:

• Real eigenvalues: All eigenvalues of a symmetric matrix are real, which simplifies their computation.
• Orthogonal eigenvectors: The eigenvectors associated with distinct eigenvalues in symmetric matrices are orthogonal, aiding in the construction of the orthogonal matrix \( P \).

These properties make symmetric matrices excellent candidates for various calculations involving projections, rotations, and optimizations. For example, in ensured stability during calculations because errors don't accumulate rapidly as they might in non-symmetric cases.
When diagonalizing symmetric matrices, the task reduces to finding real eigenvalues and orthogonal eigenvectors, making use of these attributes to simplify complex operations.