Repeated Eigenvalues 3x3

Tackling the Tricky Trio: Understanding and Solving 3x3 Matrices with Repeated Eigenvalues

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, crucial for understanding the behavior of linear transformations. While finding eigenvalues and eigenvectors for a 3x3 matrix is generally straightforward when eigenvalues are distinct, the situation becomes significantly more nuanced when we encounter repeated eigenvalues. This article delves into the complexities of solving 3x3 matrices possessing repeated eigenvalues, addressing common challenges and offering practical solutions. The ability to handle repeated eigenvalues is essential in various applications, including solving systems of differential equations, analyzing dynamical systems, and understanding the stability of physical systems.

1. Understanding the Problem: Repeated Eigenvalues in 3x3 Matrices

A 3x3 matrix A has three eigenvalues, which are the roots of its characteristic polynomial, det(A - λI) = 0, where λ represents the eigenvalues and I is the identity matrix. When an eigenvalue is repeated, we say it has algebraic multiplicity greater than 1. This repetition introduces complications because it doesn't automatically guarantee the existence of linearly independent eigenvectors associated with that eigenvalue. The geometric multiplicity (the number of linearly independent eigenvectors associated with a given eigenvalue) can be less than the algebraic multiplicity. This difference is the crux of the challenge.

2. Determining Algebraic and Geometric Multiplicity

The algebraic multiplicity of an eigenvalue is simply its multiplicity as a root of the characteristic polynomial. The geometric multiplicity, however, requires more investigation. It's determined by the dimension of the eigenspace associated with the repeated eigenvalue. The eigenspace is the null space of (A - λI), where λ is the repeated eigenvalue.

Example:

Let's consider the matrix:

A = [[2, 1, 0],
[0, 2, 0],
[0, 0, 3]]

The characteristic polynomial is (2-λ)²(3-λ) = 0. This gives eigenvalues λ₁ = 2 (algebraic multiplicity 2) and λ₂ = 3 (algebraic multiplicity 1).

For λ₁ = 2, (A - 2I) = [[0, 1, 0], [0, 0, 0], [0, 0, 1]]. Row reduction shows only one linearly independent eigenvector can be found. Therefore, the geometric multiplicity of λ₁ = 2 is 1.

For λ₂ = 3, (A - 3I) = [[-1, 1, 0], [0, -1, 0], [0, 0, 0]]. This yields one linearly independent eigenvector. The geometric multiplicity of λ₂ = 3 is 1.

3. Finding Eigenvectors for Repeated Eigenvalues

When the geometric multiplicity is less than the algebraic multiplicity, we cannot find a full set of linearly independent eigenvectors using the standard method. This necessitates the use of generalized eigenvectors.

Finding Generalized Eigenvectors:

To find generalized eigenvectors, we solve (A - λI)v = w, where λ is the repeated eigenvalue, v is the generalized eigenvector, and w is an eigenvector corresponding to λ. We repeat this process until we find a complete set of linearly independent vectors.

Example (Continuing from above):

For λ₁ = 2, we found one eigenvector, say w = [1, 0, 0]ᵀ. To find a generalized eigenvector v, we solve (A - 2I)v = w:

[[0, 1, 0],
[0, 0, 0],
[0, 0, 1]] v = [1, 0, 0]ᵀ

This gives v = [a, 1, 0]ᵀ, where 'a' is arbitrary. Let's choose a = 0, so v = [0, 1, 0]ᵀ. Now we have two linearly independent vectors, w and v, associated with λ₁ = 2.

4. Diagonalization (or Jordan Normal Form)

If the geometric multiplicity equals the algebraic multiplicity for all eigenvalues, the matrix is diagonalizable. We can form a matrix P with the eigenvectors as columns and a diagonal matrix D with the eigenvalues on the diagonal. Then A = PDP⁻¹.

If the geometric multiplicity is less than the algebraic multiplicity for any eigenvalue, the matrix is not diagonalizable. Instead, it can be transformed into its Jordan Normal Form (JNF). The JNF is a block diagonal matrix where each block corresponds to an eigenvalue and its associated generalized eigenvectors.

5. Applications and Significance

The ability to handle repeated eigenvalues is critical in various applications. For example, in solving systems of linear differential equations, repeated eigenvalues determine the form of the solution, influencing whether the system exhibits exponential growth, decay, or oscillatory behavior. In mechanics, repeated eigenvalues can indicate degeneracy in the system's normal modes of vibration. Understanding the nuances of repeated eigenvalues is therefore essential for accurate analysis and prediction in these and other fields.

Summary

Solving 3x3 matrices with repeated eigenvalues presents unique challenges that require understanding both algebraic and geometric multiplicity. When the geometric multiplicity is less than the algebraic multiplicity, the use of generalized eigenvectors becomes necessary to construct a complete set of linearly independent vectors. This allows us to either diagonalize the matrix (if possible) or transform it into its Jordan Normal Form. Mastering this process is vital for tackling problems in diverse fields where eigenvalue analysis plays a crucial role.

FAQs

1. What if I have a 3x3 matrix with three identical eigenvalues? Even with three identical eigenvalues, the geometric multiplicity might be less than 3. You'll need to investigate the eigenspace and potentially find generalized eigenvectors.

2. Can a 3x3 matrix have only one eigenvalue? Yes, but its algebraic multiplicity would be 3. The geometric multiplicity could be 1, 2, or 3.

3. How do I determine if a matrix is diagonalizable? A matrix is diagonalizable if and only if the geometric multiplicity of each eigenvalue equals its algebraic multiplicity.

4. What is the significance of the Jordan Normal Form? The Jordan Normal Form provides a canonical representation of a matrix, even when it's not diagonalizable. It simplifies calculations involving the matrix and is essential in solving systems of differential equations with repeated eigenvalues.

5. Are there software tools to assist with eigenvalue calculations? Yes, many computational software packages like MATLAB, Python's NumPy and SciPy, and Wolfram Mathematica offer functions for efficiently calculating eigenvalues and eigenvectors, including handling cases with repeated eigenvalues. These tools can greatly simplify the process and reduce the risk of calculation errors.

Search Results:

Systems of ODEs, Real Repeated Eigenvalues, 3 by 3 - BAI … How to solve systems of ordinary differential equations, using eigenvalues, real repeated eigenvalues (3 by 3 matrix) worked-out example problem. Here we will solve a system of three ODEs that have real repeated eigenvalues.

Lecture 26: Continuation: Repeated Real Eigenvalues Topics covered: Continuation: Repeated Real Eigenvalues, Complex Eigenvalues. Instructor/speaker: Prof. Arthur Mattuck

How to find the eigenvalues with repeated eigenvectors of this I know how to find the eigenvalues however for a 3x3 matrix, it's so complicated and confusing to do. So I start by writing it like this: $\begin{bmatrix}3-λ&1&1\\1&3-λ&1\\1&1&3-λ\end{bmatrix}$ and then I figure out what lambda is by finding it's determinate.

linear algebra - Finding Eigenvectors with repeated Eigenvalues ... I have a matrix $A = \left(\begin{matrix} -5 & -6 & 3\\3 & 4 & -3\\0 & 0 & -2\end{matrix}\right)$ for which I am trying to find the Eigenvalues and Eigenvectors. In this case, I have repeated Eigenvalues of $\lambda_1 = \lambda_2 = -2$ and $\lambda_3 = 1$.

Section 3.5: Repeated eigenvalues - CNRS A matrix A with two repeated eigenvalues can have: two linearly independent eigenvectors, if A = 0 0 . one linearly independent eigenvector, if A 6= 0 0 . The form and behavior of the solutions of x0= Ax is different according to these two situations. Example: Show that A = 1=2 0 0 1=2 and B = 1=2 1 0 1=2 have one repeated eigenvalue . Find ...

All tricks to find eigenvalues in - Mathematics Stack Exchange 19 Apr 2021 · Other methods exist, e.g. we know that, given that we have a 3x3 matrix with a repeated eigenvalue, the following equation system holds: $$ \left|\matrix{\text{tr}(A)= 2\lambda_1 + \lambda_2 \\ \det(A)= \lambda_1^2 \lambda_2 } \right| $$

Repeated Eigenvalues Symmetric Matrices - The University of … It follows, in considering the case of repeated eigenvalues, that the key problem is whether or not there are still n linearly independent eigenvectors for an n×n matrix. We shall now consider two 3×3 cases as illustrations.

6.8 Constant-Coefficient Homogeneous Systems: Repeated Eigenvalues In this section, we explore solutions to the homogeneous system with constant coefficients when the eigenvalues of the coefficient matrix are repeated.

$3$-by-$3$ Real Matrices with Repeated Eigenvalues Prove that for 3 × 3 3 × 3 matrices with repeated eigenvalues, all eigenvalues are real. Prove that if two eigenvalues of 3 × 3 3 × 3 are complex conjugate, then in some real basis, it takes the form ⎡⎣⎢ a −b 0 b a 0 0 0 λ⎤⎦⎥ [a b 0 − b a 0 0 0 λ].

Symmetric3 3matrices withrepeatedeigenvalues - dtrx In this article, some relations between 3 3 the matrix elements of such a symmetric matrix with a repeated eigenvalue are presented, which reduce the number of degrees of freedom of the matrix from 6 to 4. Such a matrix has 6 independent entries: 3 diagonal elements ( s22) and 3 off-s01; s02; diagonal elements ( s12).

Jordan form of 3 x 3 repeated eigenvalue - Mathematics Stack Exchange 3 May 2021 · Take the last vector $v_3$ to be any vector that is not an eigenvector for $\lambda=2$, then take $v_2=(A-2I)v_3$ (which must be an eigenvector since $\operatorname{im}(A-2I)\subset\ker(A-2I)$), and finally take $v_1$ to be an eigenvalue linearly independent of $v_2$. For instance take $v_3=(1,0,0)$, $v_2=(-1,1,2)$, and $v_1=(1,0,1)$.

3.7: Multiple Eigenvalues - Mathematics LibreTexts 24 Feb 2025 · Let us restate the theorem about real eigenvalues. In the following theorem we will repeat eigenvalues according to (algebraic) multiplicity. So for the above matrix A, we would say that it has eigenvalues 3 and 3. Take →x = P→x .

System of 3 variable differential equations with 3 repeating eigen ... 7 Jun 2018 · Here's a guide on how to deal with repeated eigenvalues. The idea is to use generalized eigenvectors such that $$ \begin{aligned} (\textbf{A}-\lambda\textbf{I})\vec{v}_1 &= \vec{0} \\ (\textbf{A}-\lambda\textbf{I})\vec{v}_2 &= \vec{v}_1 \\ (\textbf{A}-\lambda\textbf{I})\vec{v}_3 &= \vec{v}_2 \end{aligned} \tag{1} $$

Notes on repeated eigenvalues, complex eigenvalues, and the … Repeated eigenvalues: When the algebraic multiplicity k of an eigenvalue λ of A (the number of times λ occurs as a root of the characteristic polynomial) is greater than 1, we usually are not able to find k linearly independent eigenvectors

On a linear $3\times 3$ system of differential equations with repeated ... 22 Oct 2014 · When you have a triple eigenvalue and 1 single eigenvector, you have to find two eigenvectors $\mu, \text{ and }\rho,$which satisfy $$(A-\lambda I)\vec{\rho} = \vec{\eta}$$ $$\text{and}$$ $$(A-\lambda I)\vec{\mu} = \vec{\rho}.$$Then, the solution will be $$\frac{1}{2}t^2e^{\lambda t}\vec{\eta}+te^{\lambda t}\vec{\rho}+e^{\lambda t}\vec{\mu}.$$

Chapter 7 7.8 Repeated Eigenvalues - University of Kansas So, use (3) to compute another solution x(3). We proceed to solve the equation (A − rI )η = ξ. Read Example 1, 2 (They are helpful).

Repeated Eigenvalues Symmetric Matrices - Imperial College … It follows, in considering the case of repeated eigenvalues, that the key problem is whether or not there are still n linearly independent eigenvectors for an n×n matrix. We shall now consider two 3 × 3 examples as illustrations.

18.03SCF11 text: Repeated Eigenvalues - MIT OpenCourseWare We say an eigenvalue λ1 of A is repeated if it is a multiple root of the char acteristic equation of A; in our case, as this is a quadratic equation, the only possible case is when λ1 is a double real root. We need to find two linearly independent solutions …

10.5: Repeated Eigenvalues with One Eigenvector 24 May 2024 · Therefore, $\lambda=2$ is a repeated eigenvalue. The associated eigenvector is found from $-v_{1}-v_{2}=0$ , or $v_{2}=-v_{1} ;$ and normalizing with $v_{1}=1$ , we have \[\lambda=2, \quad \mathrm{v}=\left(\begin{array}{r} 1 \\ -1 \end{array}\right) \nonumber \]

Differential Equations - Repeated Eigenvalues - Pauls Online … 16 Nov 2022 · In this section we will solve systems of two linear differential equations in which the eigenvalues are real repeated (double in this case) numbers. This will include deriving a second linearly independent solution that we will need to form the general solution to the system.