How To Find Eigenvalues Given Eigenvectors

How to Find Eigenvalues Given Eigenvectors

Finding eigenvalues from eigenvectors might seem counterintuitive at first. After all, we typically find eigenvectors after we've calculated the eigenvalues. However, under specific circumstances, it's possible to determine the eigenvalues knowing only the eigenvectors and some information about the matrix. This isn't a direct, universally applicable method like calculating eigenvalues from the characteristic equation, but rather a process of deduction and leveraging the properties of eigenvectors and eigenvalues. This article explores various scenarios where this is possible and explains the procedures involved.

Understanding the Eigenvalue-Eigenvector Relationship

Before diving into the methods, let's revisit the fundamental relationship between eigenvalues (λ) and eigenvectors (v) of a square matrix A:

A v = λ v

This equation states that when a matrix A acts on an eigenvector v, the result is simply a scalar multiple (λ) of the same eigenvector. This scalar multiple is the eigenvalue. This seemingly simple equation forms the basis of all eigenvalue and eigenvector calculations.

Scenarios Where Eigenvalues Can Be Partially Determined

It's crucial to understand that we can't always determine the eigenvalues knowing only the eigenvectors. We need additional information. Here are the scenarios where it becomes feasible:

1. Knowing the Matrix A and a Set of Eigenvectors

This is the most straightforward scenario. If you have the matrix A and a set of linearly independent eigenvectors, you can find the corresponding eigenvalues by directly applying the defining equation:

A v = λ v

Solving for λ:

λ = (A v) / v (element-wise division is not generally possible; see below)

Important Note: The above equation is a simplification. Because vectors don't divide like scalars, we must perform this calculation component-wise. We solve for λ by picking a single non-zero component from the vector Av and the corresponding component from v. Let's illustrate with an example:

Let's say:

• Matrix A = [[2, 1], [1, 2]]
• Eigenvector v = [1, 1]

Then:

Av = [[2, 1], [1, 2]] * [1, 1] = [3, 3]

Now, choose a component, let's say the first component. We have:

3 = λ * 1

Therefore, λ = 3

To verify this, let's try another eigenvector:

Let's assume another eigenvector v2 = [1, -1]

Av2 = [[2, 1], [1, 2]] * [1, -1] = [1, -1]

Choosing the first component again:

1 = λ * 1

Therefore, λ = 1

Thus, for matrix A, we have eigenvalues 3 and 1.

Caution: This method only works if you already possess the matrix A and a complete set of linearly independent eigenvectors. If the eigenvectors are not linearly independent, this process becomes unreliable.

2. Utilizing the Trace and Determinant of a Matrix (for 2x2 Matrices)

For 2x2 matrices, we can leverage the relationship between the trace (sum of the diagonal elements) and the determinant to infer eigenvalues if we know the eigenvectors and some additional information. Let’s assume a 2x2 matrix A with eigenvalues λ₁ and λ₂. We know that:

• Trace(A) = λ₁ + λ₂
• Determinant(A) = λ₁ * λ₂

If we know one eigenvalue (e.g., from the method above), we can easily compute the other. If we only know the eigenvectors, this method requires further information about the matrix, such as a single element or the trace/determinant.

3. Exploiting the Properties of Symmetric Matrices

Symmetric matrices possess some unique properties concerning their eigenvectors and eigenvalues. Their eigenvectors corresponding to distinct eigenvalues are orthogonal. This orthogonality can help in deducing information about the eigenvalues even with limited information. However, this method is generally more complex and often requires additional constraints. For example, if you know the eigenvectors are orthogonal and you know one eigenvalue, you may be able to determine others based on their dot products being zero.

4. Using Special Matrix Structures

Certain types of matrices, like diagonal or triangular matrices, have eigenvalues directly visible on their diagonals. If you know the eigenvectors and you know the matrix is of one of these special types, you can directly read off the eigenvalues from the diagonal. However, this relies on prior knowledge about the structure of the matrix.

Limitations and Challenges

While we can infer eigenvalues under specific conditions, this is not a general replacement for standard eigenvalue calculation methods. Several key limitations exist:

• Information Requirement: You invariably need supplementary information beyond just the eigenvectors. This could be the matrix itself, properties of the matrix (trace, determinant), or knowledge about the matrix's structure.
• Linear Independence: The eigenvectors must be linearly independent for reliable calculations. If they're linearly dependent, the method will be prone to errors and inconsistencies.
• Computational Complexity: Even in favorable scenarios, extracting eigenvalues from eigenvectors can be computationally more involved than using traditional methods like the characteristic polynomial. The computational complexity depends heavily on the dimension of the matrix and the available information.
• Ambiguity: In some situations, there might be multiple possible sets of eigenvalues consistent with the given eigenvectors and additional information.

Advanced Techniques and Considerations

For higher-dimensional matrices or more complex scenarios, more sophisticated linear algebra techniques become necessary. These could include:

• Singular Value Decomposition (SVD): SVD can be used to analyze the relationships between eigenvectors and eigenvalues, providing further insight in certain circumstances.
• Numerical Methods: Iterative numerical methods, like the power iteration method, might help to approximate eigenvalues, though they require an initial guess and may not always converge to the desired result.

Conclusion: A Powerful but Conditional Method

Determining eigenvalues from eigenvectors is a fascinating area of linear algebra. While it's not a universally applicable method like solving the characteristic equation, it highlights the rich interconnectedness between eigenvalues and eigenvectors. The ability to deduce eigenvalues under specific circumstances enhances our understanding of the eigen-decomposition process. Always carefully consider the available information and the limitations of the method before attempting this approach. Remember that the standard methods of finding eigenvalues through the characteristic polynomial remain the most robust and widely applicable technique. This method should be seen as a complementary tool rather than a replacement for the established methods.