Problem 29 - 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 \(S^{-1} A S=J\). \(A=\left[\begin{array}{rrrr}2 & -4 & 2 & 2 \\ -2 & 0 & 1 & 3 \\ -2 & -2 & 3 & 3 \\ -2 & -6 & 3 & 7\end{array}\right] .\) [The characteristic polynomial is \(\left.p(\lambda)=(2-\lambda)^{2}(4-\lambda)^{2} .\right]\)

Step-by-Step Solution

Answer

The Jordan canonical form of the matrix A is: \[ J = \left[\begin{array}{cccc} 2 & 1 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 4 & 1\\ 0 & 0 & 0 & 4 \end{array}\right] \] The invertible matrix S for which \( S^{-1} A S = J\) is: \[ S = \left[\begin{array}{cccc} 6 & 6 & 2 & 1\\ 2 & 2 & 1 & 0\\ 4 & 0 & 0 & -1\\ 0 & 2 & 0 & 1 \end{array}\right] \]

Step 1: Compute the Eigenvalues

We are given the characteristic polynomial: \( p(\lambda) = (2-\lambda)^2(4-\lambda)^2 \). The eigenvalues are the roots of this polynomial, which are \(\lambda_1 = 2\) and \(\lambda_2 = 4\). Both eigenvalues have algebraic multiplicities of 2.

Step 2: Determine Eigenvectors and Generalized Eigenvectors

For each eigenvalue, we need to find eigenvectors and generalized eigenvectors. For \(\lambda_1=2\): Solve the equation \((A-\lambda_1 I) v_1 = 0\) for eigenvector \(v_1\). \[ \left[\begin{array}{cccc} 0 & -4 & 2 & 2\\ -2 & -2 & 1 & 3\\ -2 & -2 & 1 & 3\\ -2 & -6 & 3 & 5 \end{array}\right]v_1 = 0 \] Row reducing this matrix, we get the following matrix: \[ \left[\begin{array}{cccc} 1 & 3 & -2 & -4\\ 0 & -4 & 2 & 2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right] \] This leads to the eigenvector \(v_1 = \left[\begin{array}{c} 6 \\ 2 \\ 4 \\ 0 \end{array}\right]\). Now, we need to find a generalized eigenvector for \(\lambda_1 = 2\). We solve the equation \((A -\lambda_1 I) w_1 = v_1\). \[ \left[\begin{array}{cccc} 0 & -4 & 2 & 2\\ -2 & -2 & 1 & 3\\ -2 & -2 & 1 & 3\\ -2 & -6 & 3 & 5 \end{array}\right]w_1 = \left[\begin{array}{c} 6 \\ 2 \\ 4 \\ 0 \end{array}\right] \] Row reducing this matrix, we get the following matrix: \[ \left[\begin{array}{cccc} 1 & 3 & -2 & -4\\ 0 & -4 & 2 & 2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right] \] This gives us the generalized eigenvector \(w_1 = \left[\begin{array}{c} 6 \\ 2 \\ 0 \\ 2 \end{array}\right]\). For \(\lambda_2=4\): Solve the equation \((A-\lambda_2 I) v_2 = 0\) for eigenvector \(v_2\). \[ \left[\begin{array}{cccc} -2 & -4 & 2 & 2\\ -2 & -4 & 1 & 3\\ -2 & -2 & -1 & 3\\ -2 & -6 & 3 & 3 \end{array}\right]v_2 = 0 \] Row reducing this matrix, we get the following matrix: \[ \left[\begin{array}{cccc} 1 & 2 & -1 & -1\\ 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right] \] This leads to the eigenvector \(v_2 = \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right]\). Now, we need to find a generalized eigenvector for \(\lambda_2 = 4\). We solve the equation \((A -\lambda_2 I) w_2 = v_2\). \[ \left[\begin{array}{cccc} -2 & -4 & 2 & 2\\ -2 & -4 & 1 & 3\\ -2 & -2 & -1 & 3\\ -2 & -6 & 3 & 3 \end{array}\right]w_2 = \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] \] Row reducing this matrix, we get the following matrix: \[ \left[\begin{array}{cccc} 1 & 2 & -1 & -1\\ 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right] \] This gives us the generalized eigenvector \(w_2 = \left[\begin{array}{c} 1 \\ 0 \\ -1 \\ 1 \end{array}\right]\).

Step 3: Create Jordan Chains

We have two Jordan chains, one for each eigenvalue. Chain for \(\lambda_1 = 2\): \(v_1 = \left[\begin{array}{c} 6 \\ 2 \\ 4 \\ 0 \end{array}\right]\), \(w_1 = \left[\begin{array}{c} 6 \\ 2 \\ 0 \\ 2 \end{array}\right]\) Chain for \(\lambda_2 = 4\): \(v_2 = \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right]\), \(w_2 = \left[\begin{array}{c} 1 \\ 0 \\ -1 \\ 1 \end{array}\right]\)

Step 4: Form the Invertible Matrix S

Using the eigenvectors and generalized eigenvectors, we form the invertible matrix S. \[ S = \left[\begin{array}{cccc} 6 & 6 & 2 & 1\\ 2 & 2 & 1 & 0\\ 4 & 0 & 0 & -1\\ 0 & 2 & 0 & 1 \end{array}\right] \]

Step 5: Compute the Jordan Canonical Form J

We use the relation \( S^{-1} A S = J\) to find the Jordan canonical form J. Compute \( S^{-1} A S \) using the given matrix A and invertible matrix S calculated above: \[ J = \left[\begin{array}{cccc} 2 & 1 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 4 & 1\\ 0 & 0 & 0 & 4 \end{array}\right] \] So the Jordan canonical form of the matrix A is: \[ J = \left[\begin{array}{cccc} 2 & 1 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 4 & 1\\ 0 & 0 & 0 & 4 \end{array}\right] \]

Key Concepts

Understanding Eigenvalues

Eigenvalues are critical components in the field of linear algebra, fundamental for understanding the behavior of matrices. When we talk about eigenvalues, we refer to the special numbers associated with a matrix that tell us about its intrinsic properties. These numbers are derived from the characteristic polynomial of the matrix. By solving the equation \( (A - \lambda I) = 0 \), where \( A \) is the matrix in question and \( I \) is the identity matrix, we can find the eigenvalues, \( \lambda \).
In our exercise, the characteristic polynomial given is \( (2-\lambda)^2(4-\lambda)^2 \), providing two distinct eigenvalues, \( 2 \) and \( 4 \), each with a multiplicity of 2. These eigenvalues help inform the construction of the Jordan canonical form, allowing us to understand how the matrix can be transformed or simplified.

Unpacking Eigenvectors

Once eigenvalues are determined, the next step is to find the corresponding eigenvectors. An eigenvector for an eigenvalue \( \lambda \) is a non-zero vector \( v \) such that \( Av = \lambda v \), where \( A \) is a square matrix. In simpler terms, this vector is unaffected in direction by the matrix transformation, only changing in magnitude according to its eigenvalue.
For our specific eigenvalues \( \lambda_1 = 2 \) and \( \lambda_2 = 4 \), we solve \( (A - \lambda I)v = 0 \) to find their respective eigenvectors. An eigenvector provides a direction in which the matrix behaves as a simple scaling transformation, offering insight into the structure of the matrix.

Generalized Eigenvectors

Generalized eigenvectors extend the notion of eigenvectors, especially important when dealing with defective matrices where sufficient eigenvectors for a complete basis might be lacking. They are vectors that satisfy \( (A - \lambda I)^k v = 0 \) for some smallest positive integer \( k \). This equation maintains solutions even when the matrix does not have enough independent eigenvectors.
In our example, alongside standard eigenvectors for \( \lambda_1 = 2 \) and \( \lambda_2 = 4 \), generalized eigenvectors were found. These vectors provide deeper insight into the matrix structure, crucial for forming the Jordan chains that help construct the matrix's Jordan canonical form.

Invertible Matrix Insight

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse—the matrix \( S \) such that \( S^{-1}AS = J \), where \( J \) is the Jordan canonical form. The existence of this inverse is crucial for transforming a matrix to its canonical form.
In our task, using the eigenvectors and generalized eigenvectors, we formed such an invertible matrix \( S \). This matrix plays a pivotal role in simplifying the original matrix \( A \) to its Jordan form. Ensuring that this matrix is invertible is essential for validating the transformation process, confirming that \( S^{-1} \) exists.

Characteristic Polynomial Explained

The characteristic polynomial of a matrix is a polynomial which is invariant under matrix similarity and has the matrix's eigenvalues as roots. It is computed from the determinant of \( (A - \lambda I) \), where \( \lambda \) is a scalar and \( I \) the identity matrix. This polynomial is integral to discovering eigenvalues, serving as the starting point in matrix analysis.
For our matrix \( A \), the characteristic polynomial is \( (2-\lambda)^2(4-\lambda)^2 \), directly pointing to the eigenvalues \( 2 \) and \( 4 \). The multiplicity of these roots in the polynomial indicates how many times each eigenvalue appears as a solution to the characteristic equation, influencing the construction of its Jordan canonical form.