Problem 18 - Differential Equations and Linear Algebra

Question

Find the Jordan canonical form \(J\) for the matrix \(A_{1}\) and determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{rr}-1 & 0 \\ 0 & -2\end{array}\right]\).

Step-by-Step Solution

Answer

The Jordan canonical form \(J\) of the given matrix \(A\) is: \(J = \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}\) The invertible matrix \(S\) associated with \(A\) is simply the identity matrix: \(S = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) We have verified that \(S^{-1} A S = J\), as required.

Step 1: Find eigenvalues of the matrix A

First, we need to find the eigenvalues for the given matrix A. To find the eigenvalues, we must compute the characteristic polynomial of A, which is the determinant of (A - λI), where λ is the eigenvalue and I is the identity matrix. A - λI = \(\begin{bmatrix} -1-λ & 0 \\ 0 & -2-λ \end{bmatrix}\) The determinant of (A - λI) is: \((-1-λ)(-2-λ)\) Setting the determinant to zero, we get the eigenvalues: \((-1-λ)(-2-λ) = 0\) The eigenvalues are: λ₁ = -1 λ₂ = -2

Step 2: Find the eigenvectors associated with each eigenvalue

Now we will find the eigenvectors corresponding to each eigenvalue: For λ₁ = -1, we have: (A - λ₁I) = \(\begin{bmatrix} -1-(-1) & 0 \\ 0 & -2-(-1) \end{bmatrix}\) = \(\begin{bmatrix} 0 & 0 \\ 0 & -1 \end{bmatrix}\) Using row reduction, we get: v₁ = \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\) For λ₂ = -2, we have: (A - λ₂I) = \(\begin{bmatrix} -1-(-2) & 0 \\ 0 & -2-(-2) \end{bmatrix}\) = \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\) Using row reduction, we get: v₂ = \(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

Step 3: Form the invertible matrix S with eigenvectors

Now that we have our eigenvectors, we can form the invertible matrix S: \(S=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

Step 4: Calculate the Jordan canonical form J

Since A is already a diagonal matrix, it is also the Jordan canonical form. \(J=\begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}\) Now, we need to verify that \(S^{-1}AS=J\). Since S is the identity matrix, its inverse is also the identity matrix: \(S^{-1}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) Now, let's multiply the matrices and verify: \(S^{-1}AS=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}= \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix}\) So, we have verified that: \(S^{-1}AS=J\)

Key Concepts

Eigenvalues

Eigenvalues are special numbers associated with a matrix that offer significant insight into the matrix's properties. To find the eigenvalues, you begin by solving the characteristic equation, which is determined by setting the determinant of the matrix subtracted by \(\lambda I\) to zero. Here, \(\lambda\) represents the eigenvalue and \(I\) is the identity matrix.

• The characteristic polynomial is formed by \(\text{det}(A - \lambda I) = 0\).
• Solving this polynomial yields the eigenvalues.
• In the given exercise, the eigenvalues are \(\lambda_1 = -1\) and \(\lambda_2 = -2\).

By determining these special values, we understand how the transformation represented by matrix \(A\) affects vectors in the space.

Eigenvectors

Eigenvectors are vectors associated with a particular eigenvalue of a matrix. They remain on the same line when transformed by the matrix. To find an eigenvector corresponding to an eigenvalue, you perform the following steps:

• Substitute an eigenvalue \(\lambda\) into \(A - \lambda I\) to form a new matrix.
• Use row reduction to find the null space of this matrix.
• The solution to this set of linear equations is the eigenvector.

In our exercise, for \(\lambda_1 = -1\), the eigenvector \(v_1\) is \(\begin{bmatrix} 1 \ 0 \end{bmatrix}\). Similarly, for \(\lambda_2 = -2\), the eigenvector \(v_2\) is \(\begin{bmatrix} 0 \ 1 \end{bmatrix}\). These vectors help identify directions in which transformation by \(A\) only stretches or compresses.

Diagonalization

Diagonalization is a process of finding a diagonal matrix that represents the same linear transformation as a given square matrix. If such a representation is possible, a matrix is said to be diagonalizable. The steps involved in diagonalization are:

• Find eigenvalues and corresponding eigenvectors of the matrix.
• Form a matrix \(S\) with the eigenvectors as columns.
• Construct a diagonal matrix \(D\) with the eigenvalues on the diagonal.
• Verify that \(A = SDS^{-1}\).

In the exercise, matrix \(A\) is already diagonal, simplifying the process. The matrix \(S\) becomes the identity matrix, and the Jordan form \(J\) is equivalent to \(A\), confirming the diagonal nature of the transformation.

Matrix Theory

Matrix Theory is a fundamental area of mathematics with applications across science and engineering. It deals with the study of matrices and their properties. Some key concepts include:

• Matrix operations such as addition, multiplication, and inversion.
• The role of identity matrix and zero matrix in linear transformations.
• Determinants and their use in solving systems of equations and finding eigenvalues.
• Special matrices like diagonal and Jordan forms that aid in simplifying calculations.

Understanding these concepts is crucial, as they form the backbone for advanced topics like transforming matrices into simpler forms (like diagonal or Jordan form), which helps in solving linear equations, analyzing system stability, and more.