Eigenvalue and Eigenvector: Insights into Linear Transformations

Understand eigenvalues and eigenvectors, scalars and vectors that provide significant insight into the properties of linear transformations represented by matrices.

In the field of linear algebra, eigenvalues and eigenvectors are fundamental concepts that provide deep insights into the properties of linear transformations.

Definition

An eigenvalue is a scalar \(\lambda\) associated with a given square matrix \(A\), such that when the matrix multiplies a non-zero vector (called an eigenvector) \(v\), the result is a scalar multiple of \(v\). Mathematically, this relationship can be expressed as:

$$ A v = \lambda v $$

Here:

• \(A\) is a \(n \times n\) matrix.
• \(v\) is a non-zero vector, also known as an eigenvector, in \(\mathbb{R}^n\).
• \(\lambda\) is a scalar known as an eigenvalue.

Types of Eigenvalues and Eigenvectors

Real and Complex Eigenvalues

• Real Eigenvalues: If the eigenvalue \(\lambda\) is a real number, the eigenvector associated with it is also in the real domain.
• Complex Eigenvalues: If \(\lambda\) is a complex number, the eigenvector \(v\) will also be in the complex domain.

Distinct and Repeated Eigenvalues

• Distinct Eigenvalues: Eigenvalues that are all different from each other.
• Repeated Eigenvalues: An eigenvalue that appears more than once in the eigenvalue spectrum of the matrix, known as algebraic multiplicity.

Properties and Calculation

Characteristic Equation

To find the eigenvalues of a matrix \(A\), we solve the characteristic equation:

$$ \text{det}(A - \lambda I) = 0 $$

Here:

• \(\text{det}\) denotes the determinant.
• \(I\) is the identity matrix of the same order as \(A\).
• \(\lambda\) represents the eigenvalues.

Eigenvectors

After determining the eigenvalues, we can find the eigenvectors by solving:

$$ (A - \lambda I)v = 0 $$

This forms a system of linear equations that can be solved to obtain the eigenvectors.

Examples

Example 1: 2x2 Matrix

Consider the matrix:

$$ A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} $$

• Step 1: Find the characteristic equation.

$$ \text{det}(A - \lambda I) = \begin{vmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{vmatrix} $$

$$ (4 - \lambda)(3 - \lambda) - 2 \cdot 1 = \lambda^2 - 7\lambda + 10 = 0 $$

• Step 2: Solve for \(\lambda\).

$$ \lambda = 2 \text{ or } 5 $$

• Step 3: Find the eigenvectors for each \(\lambda\).

For \(\lambda = 2\), solve:

$$ (A - 2I)v = 0 \Rightarrow \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$

One solution is \(v_1 = \begin{pmatrix} -0.5 \ 1 \end{pmatrix}\).

For \(\lambda = 5\), solve:

$$ (A - 5I)v = 0 \Rightarrow \begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$

One solution is \(v_2 = \begin{pmatrix} 1 \ 1 \end{pmatrix}\).

Historical Context

Origin

The concept of eigenvalues and eigenvectors dates back to the 18th century with the work of Leonhard Euler and later formalized by Augustin-Louis Cauchy. The term “eigenvalue” (origin: German “Eigenwert”) was introduced by David Hilbert in the early 20th century.

Applicability

Eigenvalues and eigenvectors have applications in various fields including:

• Physics: Vibration analysis and quantum mechanics.
• Engineering: Control systems and structural analysis.
• Economics: Markov chains and dynamic systems.
• Computer Science: Principal Component Analysis (PCA) and Google’s PageRank algorithm.

Comparisons

Similar Terms

• Singular Value Decomposition (SVD): A generalization of the eigenvalue decomposition for non-square matrices.
• Characteristic Polynomial: Polynomial equation derived from the characteristic equation.

Differentiated Terms

• Eigenvalue measures the factor by which the eigenvector is scaled.
• Eigenvector indicates the direction of the transformation.

FAQs

What is the geometric interpretation of eigenvalues and eigenvectors?

Eigenvalues represent the scale factors, while eigenvectors indicate the directions along which the linear transformation acts by stretching or compressing.

Can a matrix have a zero eigenvalue?

Yes, if \(\lambda = 0\) is an eigenvalue, the matrix is said to be singular, implying it does not have an inverse.

Are eigenvalues always real?

No, eigenvalues can be complex numbers, especially for matrices with complex entries or certain real matrices like those that are not symmetric.

References

1. Strang, Gilbert. Linear Algebra and Its Applications. Brooks Cole, 2005.
2. Axler, Sheldon. Linear Algebra Done Right. Springer, 2015.
3. Stewart, James. Calculus: Early Transcendentals. Cengage Learning, 2015.

Summary

Eigenvalues and eigenvectors serve as indispensable tools in decomposition, analysis, and understanding of linear transformations. They hold significant importance across various disciplines, providing insights into the essence of matrix algebra and facilitating solutions to complex mathematical problems.

By translating matrix multiplicative operations into scalar multiplicative operations via eigenvalues and eigenvectors, complex systems become more comprehensible and manageable.