Glossary

1.3 Eigenvalues, Eigenvectors, and Diagonalization

3 min read•july 22, 2024

Eigenvalues and eigenvectors are key to understanding how matrices transform space. They reveal special directions where a matrix only stretches or shrinks vectors, without changing their orientation. This concept is crucial for solving complex problems in physics and engineering.

Diagonalization uses eigenvalues and eigenvectors to simplify matrices. By breaking down a matrix into simpler parts, we can more easily solve systems of equations and differential equations. This technique is widely used in and other advanced physics applications.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors basics

Top images from around the web for Eigenvalues and eigenvectors basics

  • Eigenvalues and eigenvectors - Wikipedia View original

    Is this image relevant?

  • Eigenvalues and eigenvectors - Wikipedia View original

    Is this image relevant?

  • Eigenvalues and eigenvectors - Wikipedia View original

    Is this image relevant?

1 of 3

Top images from around the web for Eigenvalues and eigenvectors basics

  • Eigenvalues and eigenvectors - Wikipedia View original

    Is this image relevant?

  • Eigenvalues and eigenvectors - Wikipedia View original

    Is this image relevant?

  • Eigenvalues and eigenvectors - Wikipedia View original

    Is this image relevant?

1 of 3
  • Eigenvalues (λ\lambda) scalar values that when a square matrix AA is multiplied by a non-zero vector [v](https://www.fiveableKeyTerm:v)⃗\vec{[v](https://www.fiveableKeyTerm:v)}, the result is a scalar multiple of v⃗\vec{v} (Av⃗=λv⃗A\vec{v} = \lambda\vec{v})
  • Eigenvectors (v⃗\vec{v}) non-zero vectors that when multiplied by a square matrix AA, result in a scalar multiple of themselves
    • Eigenvectors remain unchanged in direction after the represented by matrix AA
  • Geometrically, eigenvectors represent directions in which a linear transformation scales without changing direction
    • Corresponding eigenvalues represent scale factors by which eigenvectors are stretched or compressed (scaling factor, dilation)

Computation of eigenvalues and eigenvectors

  • Characteristic equation of a square matrix AA: det⁡(A−λI)=0\det(A - \lambda I) = 0, where II is the identity matrix
    • Solving characteristic equation yields eigenvalues of matrix AA
  • To find eigenvectors for an λ\lambda, solve equation (A−λI)v⃗=0⃗(A - \lambda I)\vec{v} = \vec{0} for non-zero vectors v⃗\vec{v}
    • Solutions to this equation are eigenvectors associated with eigenvalue λ\lambda
  • Other methods for computing eigenvalues and eigenvectors:
    • Power iteration method iteratively multiplies a vector by the matrix to approximate dominant eigenvalue and corresponding
    • Singular Value Decomposition (SVD) factorizes a matrix into product of three matrices, revealing eigenvalues and eigenvectors (decomposition, factorization)

Diagonalization

Matrix diagonalization process

  • A square matrix AA is diagonalizable if A=PDP−1A = PDP^{-1}, where DD is a diagonal matrix and PP is an invertible matrix
    • Diagonal entries of DD are eigenvalues of AA
    • Columns of PP are eigenvectors of AA
  • For a matrix to be diagonalizable, it must have a full set of linearly independent eigenvectors
    • Number of linearly independent eigenvectors must equal dimension of the matrix
  • Steps to diagonalize a matrix AA:
    1. Find eigenvalues of AA by solving characteristic equation
    2. For each distinct eigenvalue, find corresponding eigenvectors
    3. Form matrix PP by placing eigenvectors as columns
    4. Form diagonal matrix DD with eigenvalues as diagonal entries
    5. Verify A=PDP−1A = PDP^{-1}

Applications in differential equations

  • A system of linear differential equations in matrix form: dx⃗dt=Ax⃗\frac{d\vec{x}}{dt} = A\vec{x}, where AA is a square matrix and x⃗\vec{x} is a vector of functions
  • If AA is diagonalizable, solve the system by:
    1. Diagonalizing AA as A=PDP−1A = PDP^{-1}
    2. Introducing new variable y⃗=P−1x⃗\vec{y} = P^{-1}\vec{x}
    3. Rewriting system as dy⃗dt=Dy⃗\frac{d\vec{y}}{dt} = D\vec{y}, which decouples equations
    4. Solving each decoupled equation independently
    5. Transforming solution back to original variables using x⃗=Py⃗\vec{x} = P\vec{y}
  • Solution to the system will be a linear combination of exponential functions
    • Eigenvalues appear in exponents
    • Eigenvectors determine coefficients (weights, constants)

Key Terms to Review (16)

Algebraic Multiplicity: Algebraic multiplicity refers to the number of times a particular eigenvalue appears as a root of the characteristic polynomial of a matrix. It indicates how many linearly independent eigenvectors correspond to that eigenvalue and plays a crucial role in determining the diagonalizability of a matrix. The relationship between algebraic multiplicity and geometric multiplicity is fundamental for understanding the structure of linear transformations.
Characteristic Polynomial: The characteristic polynomial of a square matrix is a polynomial which is derived from the determinant of the matrix subtracted by a scalar multiple of the identity matrix. This polynomial plays a crucial role in determining the eigenvalues of the matrix, as the roots of the characteristic polynomial correspond to these eigenvalues. Understanding the characteristic polynomial is essential for concepts such as diagonalization, as it helps reveal properties of matrices that can simplify complex linear transformations.
Diagonalizability: Diagonalizability refers to the property of a matrix that allows it to be expressed as a product of three matrices: a diagonal matrix and two invertible matrices. This property simplifies matrix operations, especially when raising matrices to powers or solving differential equations, making it easier to analyze linear transformations associated with the matrix.
Eigenvalue: An eigenvalue is a scalar associated with a linear transformation represented by a matrix, indicating how much a corresponding eigenvector is stretched or compressed during that transformation. The relationship between eigenvalues and eigenvectors is crucial in understanding the behavior of linear transformations, especially in terms of stability and dynamics. In many applications, particularly in physics and engineering, identifying eigenvalues helps in solving systems of differential equations and optimizing functions.
Eigenvector: An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it. In simpler terms, when you multiply an eigenvector by a matrix, the result is just that eigenvector scaled by some number known as the eigenvalue. Eigenvectors are crucial in understanding how matrices can be simplified, especially when discussing concepts like diagonalization and spectral theory, where they help reveal the underlying structure of linear transformations.
Geometric Multiplicity: Geometric multiplicity refers to the number of linearly independent eigenvectors associated with a given eigenvalue of a matrix. This concept is crucial in understanding the behavior of matrices, particularly in the context of diagonalization, as it indicates how many dimensions in the vector space are spanned by the eigenvectors corresponding to that eigenvalue.
Jordan Form: Jordan Form, or Jordan Canonical Form, is a special type of matrix representation that simplifies the study of linear transformations by organizing a matrix into a block diagonal structure. This form reveals the eigenvalues and their corresponding geometric and algebraic multiplicities, making it easier to analyze the behavior of linear operators. It is particularly important when dealing with matrices that cannot be diagonalized, as it provides a way to still understand their essential properties.
Linear Transformation: A linear transformation is a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This concept plays a crucial role in various mathematical areas, allowing for simplification of complex problems, particularly in changing coordinate systems and understanding how different representations relate to each other. The transformation can often be represented using matrices, which connects it to eigenvalues and eigenvectors, revealing important properties about the transformation's effect on space.
Matrix representation: Matrix representation is a mathematical framework used to express linear transformations in terms of matrices. It provides a structured way to manipulate and analyze these transformations by translating them into numerical form, facilitating operations such as addition, scalar multiplication, and composition. This concept is crucial for understanding how vectors are transformed and how systems of equations can be solved more efficiently.
QR Algorithm: The QR algorithm is a numerical method used to compute the eigenvalues and eigenvectors of a matrix by decomposing it into an orthogonal matrix Q and an upper triangular matrix R. This algorithm is particularly important in the study of eigenvalues and diagonalization, as it provides a systematic approach to approximate these values iteratively, leading to convergence for many types of matrices.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It introduces concepts such as wave-particle duality, quantization of energy levels, and the uncertainty principle, which challenge classical mechanics and provide a framework for understanding phenomena like atomic structure and chemical reactions.
Similar Matrices: Similar matrices are two square matrices that represent the same linear transformation in different bases, meaning they can be transformed into each other by a change of basis represented by an invertible matrix. The concept of similarity connects deeply to eigenvalues and eigenvectors, as similar matrices share the same eigenvalues, which indicates that they have equivalent properties in terms of their spectral characteristics. Additionally, the ability to diagonalize a matrix relies heavily on the idea of similarity since a matrix can be diagonalized if it is similar to a diagonal matrix.
Spectral Theorem: The spectral theorem is a fundamental result in linear algebra and functional analysis that characterizes certain types of linear operators, particularly self-adjoint operators, by their eigenvalues and eigenvectors. This theorem provides a way to diagonalize these operators, allowing them to be expressed in a simpler form that reveals their underlying structure and behavior. Understanding this theorem is crucial for solving problems related to quantum mechanics and various applications in mathematical physics.
V: In the context of eigenvalues, eigenvectors, and diagonalization, the term 'v' typically represents an eigenvector associated with a specific eigenvalue of a matrix. Eigenvectors are non-zero vectors that, when multiplied by a matrix, result in a vector that is a scalar multiple of themselves. This property is crucial for understanding how matrices transform vectors and plays a significant role in simplifying complex linear transformations.
Vibration Modes: Vibration modes refer to the distinct patterns in which a system oscillates when it vibrates. Each mode corresponds to a specific frequency at which the system can resonate, leading to unique spatial distributions of displacement among its components. Understanding these modes is crucial as they are directly related to the eigenvalues and eigenvectors associated with the system's mathematical description, allowing for analysis of stability and dynamic behavior.
λ (Lambda): In the context of linear algebra, λ (lambda) is a scalar value known as an eigenvalue. It is associated with a matrix and characterizes how vectors, specifically eigenvectors, are stretched or compressed when that matrix is applied to them. When you multiply an eigenvector by its corresponding matrix, the result is a new vector that is a scaled version of the original vector, where the scaling factor is the eigenvalue λ.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide