Glossary

1.3 Eigenvalues, eigenvectors, and diagonalization

2 min read•july 25, 2024

Eigenvalues and eigenvectors are key concepts in linear algebra, crucial for understanding how matrices transform vectors. They help simplify complex calculations and reveal important properties of linear transformations, making them essential tools in various fields.

These concepts are particularly useful in quantum mechanics and classical mechanics. They allow us to solve differential equations, analyze dynamical systems, and understand the behavior of physical systems under various transformations and conditions.

Fundamentals of Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors

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  • Eigenvalues (λ\lambda) scalar values satisfy Av⃗=λv⃗A\vec{v} = \lambda\vec{v}, represent scaling of by linear transformation
  • Eigenvectors (v⃗\vec{v}) non-zero vectors satisfy Av⃗=λv⃗A\vec{v} = \lambda\vec{v}, maintain direction under linear transformation
  • Calculation process involves:
    1. Set up equation (A−λI)v⃗=0⃗(A - \lambda I)\vec{v} = \vec{0}
    2. Solve for λ\lambda using characteristic equation
    3. Find corresponding eigenvectors by solving (A−λI)v⃗=0⃗(A - \lambda I)\vec{v} = \vec{0} for each λ\lambda

Characteristic equation for eigenvalues

  • Characteristic equation defined as det(A−λI)=0det(A - \lambda I) = 0, polynomial equation in λ\lambda
  • Steps to find eigenvalues:
    1. Compute det(A−λI)det(A - \lambda I)
    2. Set determinant equal to zero
    3. Solve resulting polynomial equation
  • Multiplicity of eigenvalues includes algebraic multiplicity (number of times λ\lambda appears as root) and geometric multiplicity (dimension of eigenspace for λ\lambda)

Geometric interpretation of eigenpairs

  • Eigenvectors represent directions only scaled, not rotated, by transformation
  • Eigenvalues act as scaling factors along eigenvector directions
  • Significance in linear transformations:
    • Eigenspaces remain unchanged under transformation (invariant subspaces)
    • Eigenvectors often represent principal axes of transformation
  • Applications include principal component analysis, stress and strain analysis (mechanics), energy states and wavefunctions (quantum mechanics)

Diagonalization of matrices

  • Diagonalization process finds similar to given matrix: A=PDP−1A = PDP^{-1}, where DD is diagonal and PP is eigenvector matrix
  • Conditions for diagonalizability:
    • Matrix must have nn linearly independent eigenvectors (nn is matrix size)
    • Geometric multiplicity equals algebraic multiplicity for each eigenvalue
  • Checking diagonalizability involves computing eigenvalues and multiplicities, finding eigenvectors, checking linear independence, ensuring number of linearly independent eigenvectors equals matrix size

Applications of matrix diagonalization

  • Diagonalization process:
    1. Find eigenvalues and eigenvectors
    2. Form eigenvector matrix PP and diagonal matrix DD
    3. Verify A=PDP−1A = PDP^{-1}
  • Simplifying calculations:
    • Powers of matrices: An=PDnP−1A^n = PD^nP^{-1}
    • Matrix exponentials: eAt=PeDtP−1e^{At} = Pe^{Dt}P^{-1}
  • Applications in repeated transformations:
    • Dynamical systems: x⃗(t)=eAtx⃗(0)\vec{x}(t) = e^{At}\vec{x}(0)
    • Markov chains: long-term behavior determined by eigenvalues
  • Solving differential equations by decoupling systems of linear differential equations, finding general solutions using eigenvectors as basis

Key Terms to Review (13)

Characteristic Polynomial: The characteristic polynomial is a polynomial equation that is derived from a square matrix and is used to determine the eigenvalues of that matrix. By setting the characteristic polynomial equal to zero, we can find the eigenvalues, which are crucial in understanding the behavior of linear transformations. This polynomial encapsulates important properties of the matrix, including its eigenvalues, which are key in diagonalization and the study of matrix functions.
Diagonal Matrix: A diagonal matrix is a special type of square matrix in which all the elements outside the main diagonal are zero. This unique structure allows for simplifications in various mathematical operations, particularly when it comes to eigenvalues and eigenvectors. The significance of diagonal matrices arises prominently in the process of diagonalization, where a matrix can be expressed in a simplified form that reveals important properties of linear transformations.
Diagonalization Theorem: The diagonalization theorem states that a square matrix can be expressed in a diagonal form if it has enough linearly independent eigenvectors. This theorem is crucial because diagonal matrices are much easier to work with in computations, especially when it comes to raising the matrix to powers or finding functions of the matrix. The ability to diagonalize a matrix simplifies many problems in linear algebra and is fundamental in areas such as differential equations and quantum mechanics.
Eigenvalue: An eigenvalue is a special number associated with a linear transformation represented by a matrix, which indicates how much an eigenvector is stretched or compressed during that transformation. Eigenvalues reveal important properties about the behavior of a system and are crucial for solving differential equations, especially in the context of physical systems. They are central to understanding stability, resonance, and quantization in mechanics and quantum mechanics.
Eigenvector: An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied to it. This means that when you apply a matrix to an eigenvector, the result is simply the eigenvector scaled by a corresponding eigenvalue. Eigenvectors are crucial in understanding the behavior of linear transformations and play a key role in diagonalization and spectral theory.
Matrix Factorization: Matrix factorization is the process of decomposing a matrix into a product of two or more matrices, simplifying complex data representations and revealing underlying structures. This technique is fundamental in linear algebra, particularly in finding eigenvalues and eigenvectors, which are crucial for understanding the behavior of linear transformations and systems. In many cases, matrix factorization enables diagonalization, allowing matrices to be expressed in a simpler form that is easier to manipulate for various applications such as solving differential equations or optimizing systems.
Normal Modes: Normal modes are specific patterns of oscillation in a system where all parts of the system oscillate at the same frequency, maintaining a constant phase relationship. These modes are fundamental solutions to the equations of motion and can be understood through the framework of eigenvalues and eigenvectors, which characterize how a system responds to perturbations.
Quantum States: Quantum states are mathematical descriptions of the physical properties of a quantum system, encapsulating all the information about the system's behavior and characteristics. They are represented in a Hilbert space and can take various forms, such as wave functions or state vectors, allowing for the representation of superpositions and entanglement, which are fundamental concepts in quantum mechanics.
Similarity Transformation: A similarity transformation is a mathematical operation that transforms a matrix into another matrix that retains its eigenvalues and geometric properties, thereby preserving the essential characteristics of the system represented by the matrix. This type of transformation is often used in the context of diagonalization, where a matrix can be represented in a simpler form, making it easier to analyze and solve problems related to eigenvalues and eigenvectors.
Spectral Theorem: The spectral theorem is a fundamental result in linear algebra that states that any normal operator on a finite-dimensional inner product space can be diagonalized by an orthonormal basis of eigenvectors. This theorem connects the concepts of eigenvalues and eigenvectors to the representation of linear transformations, allowing for the analysis of observables in quantum mechanics, particularly when dealing with Hermitian operators, which represent measurable quantities. It also provides a framework for understanding the role of Dirac notation and matrix representations in describing quantum states and operators.
Symmetric matrix: A symmetric matrix is a square matrix that is equal to its transpose, meaning that the element in the i-th row and j-th column is the same as the element in the j-th row and i-th column for all indices. This property ensures that symmetric matrices have real eigenvalues and their eigenvectors corresponding to distinct eigenvalues are orthogonal, which plays a crucial role in simplifying many problems in linear algebra and its applications.
Trace: In linear algebra, the trace of a square matrix is defined as the sum of its diagonal elements. This concept connects to various mathematical applications, including eigenvalues, where the trace can provide insight into the eigenvalues of a matrix, as it equals the sum of those eigenvalues. Additionally, in quantum mechanics, the trace plays a crucial role in analyzing density matrices and mixed states, helping to quantify probabilities and expected values.
Transformation matrix: A transformation matrix is a mathematical construct used to represent linear transformations from one vector space to another. It acts on vectors to perform operations like rotation, scaling, or translation. The connection to eigenvalues and eigenvectors comes into play when we want to simplify a linear transformation by finding a suitable basis in which the matrix takes a diagonal form, making calculations easier.
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