What is the difference between eigenvalues and eigenvectors? - GeeksforGeeks (2024)

Last Updated : 28 Jun, 2024


Eigenvalues and eigenvectors are concepts that are used in Linear Algebra which is an important branch of Mathematics and which serves as the basis of many areas of Physics, Engineering and even Computer Science. The eigenvector of the matrix is the non-zero vector which either extends or reduces only using scalar factor while being operated by that matrix. The corresponding eigenvalue is the scalar which is used for describing how the eigenvector is changed, that is, scaled, during this transformation. Both form a toolkit about linear transformations as linear operators and their properties and behaviour which are crucial for deepening understanding of complex properties of a given system and solving diverse differential equations. This introduction lays the foundation on which to discuss the mathematical specifics of eigenvalues and eigenvectors as well as highlight their uses.

What is eigenvalues

Eigenvalues are the numerical values that are related to the eigenvectors in linear transform. The term Eigen is a word borrowed from the German language and is used in mathematics and is derived from the German word Eigen meaning characteristic. Therefore, these are eigenvalues, which express the amount by which eigenvectors are expanded in the direction of the eigenvector. It does not presuppose the alteration of the orientation of the vector apart from instances where the eigenvalue is negative. When the Eigenvalue is negative it’s just reversed in direction. The equation for eigenvalue is given

Av = λv


  • A is the matrix,
  • v is associated eigenvector, and
  • λ is scalar eigenvalue.

    What is eigenvectors

    Eigenvectors for square matrices are defined as non-zero vector values such that when the vectors are multiplied by the square matrices, the resultant matrix is the scaler multiple of the vector that is, we define an eigenvector for matrix A as “v” that satisfies the following condition Av = λv

    The scaler multiple λ in the above case is known as the eigenvalue of the above-being square matrix. In almost every case, we are first required to find the eigenvalues of the square matrix before we look for the eigenvectors of the matrix.

    Eigenvector Equation

    The Eigenvector equation is the equation that is used to find the eigenvector of any square matrix. The eigenvector equation is,

    Av = λv


    • A is the given square matrix,
    • v is the eigenvector of matrix A, and
    • λ is any scaler multiple.

      Difference between eigenvalues and eigenvectors





      Scalars indicating how a transformation scales

      Vectors indicating the direction of scaling

      Mathematical Representation




      Eigenvalues exist for every matrix

      Eigenvectors exist only for non-zero eigenvalues


      There can be multiple eigenvalues, some may be repeated

      Corresponding eigenvectors can form a subspace

      Dependence on Matrix

      Directly derived from the characteristic polynomial

      Derived from solving (A-xI)v = 0


      Always scalars

      Always vectors

      Geometric Interpretation

      Represents scaling factor

      Represents direction vector


      Single dimension (scalar)

      Same dimension as the original matrix (vector)

      Physical Interpretation

      Can represent frequencies, energy levels, etc.

      Can represent vibration modes, principal directions


      Generally simpler to compute

      Involves solving linear equations

      Symmetry in Real Matrices

      Eigenvalues are real for symmetric matrices

      Eigenvectors are orthogonal for symmetric matrices

      Applications of Eigenvalues

      • Stability Analysis in Dynamical Systems: Eigenvalues are used to determine the stability of equilibrium points in different dynamic systems. The eigenvalues of the Jacobian matrix of the system at an equilibrium point are all negative in real parts, the equilibrium is stable The eigenvalues of the Jacobian matrix of the system with positive real parts the equilibrium is unstable.
      • Quantum Mechanics: In quantum mechanics, eigenvalues are states of a physical system that correspond to a particular value of an operator, for example, the energy of an electron in an atom. For example, the Schrödinger equation involves searching for eigenvalues of the Hamilton operators which are equivalent to the energies of the system.
      • Vibration Analysis: Eigenvalues are employed in the estimation of the natural frequency of vibration of a mechanical structure. Natural frequencies that are quite important in attempting to counter-check any cases of resonance in structures come from the eigenvalues of the system mass and stiffness matrices.
      • Principal Component Analysis (PCA): PCA is a method that is implemented under the category of exploratory factor analysis, as a method of the reduction of variables. From the covariance matrix, the eigenvalues that define the principal components tell us the proportion of variance explained by each of the new variables. The larger eigenvalues represent features with more spread-out values in the dataset.
      • Markov Chains: In Markov chains, the measure of the distances between states is expressed via the eigenvalues of the Transition Probability matrix to describe the long-term behaviour of the dynamical system. The first eigenvalue can be the largest in the case of a stochastic matrix with the value often being 1 and others are about how fast it will converge.

      Applications of Eigenvectors

      • Principal Component Analysis (PCA): Since eigenvectors of the covariance matrix can be viewed as directions of maximum variability, they are called also principal components. Eigenvectors above are used to map the originally used matrices into a new Fourier space of lower Dimensionality.
      • Molecular Orbital Theory: Eigenvectors of the Hamiltonian matrix in the molecular orbital theory of chemistry provide the molecular orbitals. The eigenvalues are the same as the energy levels of such orbitals, while the eigenvectors define the spatial configuration of the orbitals.
      • Google PageRank Algorithm: Google uses a method called PageRank to sort the Web pages and it is defined by the eigenvector of the Web page graph’s adjacency matrix. The principal eigenvector is a contour of the random surfers’ long-term behaviour to give the importance of each of the pages.
      • Image Compression: Compression techniques involving the use of Singular Value Decomposition (SVD) work by using eigenvectors of the image matrix to reconstruct the image with lower data. The eigenfeatures equal to the largest eigenvalues contain features of an image that can be important during the determination of likeness.
      • Mechanical Vibrations and Mode Shapes: In Structural dynamics, eigenvectors are used as the mode shapes of a vibrating system. These mode shapes characterise the vibration pattern or distortion of the structure at each natural frequency, which is key data in the design and analysis of structures influenced by the dynamic load.


      In conclusion, eigenvalues and eigenvectors are core topics in linear algebra that help to comprehend linear transformations and their impact on vector spaces. Therefore, eigenvalues as scalars measure the extent of this scaling while eigenvectors as vectors identify the direction of this scaling. Its use covers areas of stability analysis, quantum mechanics, vibration analysis and also data reduction techniques such as the principal component analysis. Their long-standing descriptions enhance the knowledge of their behaviour in the systems and matrices that we come across.

      FAQs – Difference between eigenvalues and eigenvectors

      Can a matrix have complex eigenvalues?

      A matrix can have complex eigenvalues, even in cases where the matrix is not symmetric. Conjugate pairs of complex eigenvalues are useful in control theory and signal processing where many real applications are found.

      How do eigenvalues and eigenvectors relate to diagonalization?

      If it is possible to write all elements of a matrix A in the form A=PDP^(-1) where all elements of the matrix D are eigenvalues of A and all columns of the matrix P are the corresponding eigenvectors then the matrix A is said to be diagonalizable. Diagonalization proves beneficial in simplifying various other operations related to the matrices.

      What is the geometric significance of eigenvectors?

      Eigenvectors are ways of similar transformation, in terms of geometry they are directions along which a linear transformation scales the vector without rotation. This aids in comprehending the characteristics of linear transformations as well as their mapping patterns.

      Can the eigenvalues of a matrix be zero?

      Yes, a matrix can have zero as one of its eigenvalues. But, we know that if an eigenvalue λ=0, then the matrix is singular or, in other words, the inverse of the matrix does not exist and the determinant of the matrix is equal to zero.

      How are eigenvalues used in the study of differential equations?

      It has been mentioned that eigenvalues are applied to solve linear differential equations especially systems of linear differential equations. They assist in explaining the state and tendencies of solutions at different periods, particularly in the application of dynamic models.

      Previous Article

      Project Analyst Vs Project Manager: What's the Difference?

      Next Article

      What is the difference between a strong AI and a weak AI?

      Please Login to comment...

      What is the difference between eigenvalues and eigenvectors? - GeeksforGeeks (2024)


      What is the difference between eigenvalues and eigenvectors? - GeeksforGeeks? ›

      Eigenvalues are the scaler which is used to find eigenvectors whereas eigenvectors are the vectors that are used to find matrix vector transformations.

      What is the difference between eigenvalue and eigenvector? ›

      Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed.

      What are eigenvectors and eigenvalues in layman's terms? ›

      In Mathematics, an eigenvector corresponds to the real non zero eigenvalues which point in the direction stretched by the transformation whereas eigenvalue is considered as a factor by which it is stretched. In case, if the eigenvalue is negative, the direction of the transformation is negative.

      What are eigenvalues and eigenvectors simplified? ›

      Eigenvalues are special numbers associated with a matrix and eigenvectors are special vectors. A matrix A acts on vectors x like a function does, with input x and output Ax. Eigenvectors are vectors for which Ax is parallel to x.

      What is an example of an eigenvalue in real life? ›

      Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes, eigenvectors. Control theory, vibration analysis, electric circuits, advanced dynamics and quantum mechanics are just a few of the application areas.

      What do eigenvectors tell us? ›

      The eigenvectors show us the direction of our main axes (principal components) of our data. The greater the eigenvalue, the greater the variation along this axis. So the eigenvector with the largest eigenvalue corresponds to the axis with the most variance.

      What is an eigenvector in layman's terms? ›

      Eigenvector. Every vector (list of numbers) has a direction when it is plotted on XY chart. Eigenvectors are those vectors when a linear transformation (such as multiplying it to a scalar) is performed on them, their direction does not change.

      What is the purpose of eigenvalues? ›

      Eigenvalues ​​characterize important properties of linear transformations, such as whether a system of linear equations has a unique solution or not. In many applications eigenvalues ​​also describe physical properties of a mathematical model.

      Can an eigenvalue have no eigenvector? ›

      Every eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so γA(λ)≥1 γ A ( λ ) ≥ 1 .

      What is the simple application of eigenvalues and eigenvectors? ›

      Eigenvalues and eigenvectors are extensively used in power system analysis and the study of electromagnetic fields. In power systems, eigenvalues help determine the stability and transient response of the system, aiding in voltage stability analysis and fault detection.

      What is an eigenvalue for dummies? ›

      This linear transformation gets described by a matrix called the eigenvector. The points in that matrix are called eigenvalues. Think of it this way: the eigenmatrix contains a set of values for stretching or shrinking your legs. Those stretching or shrinking values are eigenvalues.

      What are the advantages and disadvantages of eigenvalues and eigenvectors? ›

      • Advantages:
      • Eigenvectors allow you to solve any N dimensional problems in N Linear time which is computational optimal.
      • Eigenvalues are the magnitudes of the principle (eigen) vectors.
      • Disadvantages:
      • You have the calculate them for each problem unless you have a degenerate special case and they are given.
      Aug 3, 2018

      Does every matrix have eigenvalues? ›

      Every real matrix has an eigenvalue, but it may be complex. In fact, a field K is algebraically closed iff every matrix with entries in K has an eigenvalue. You can use the companion matrix to prove one direction.

      What is the practical use of eigenvalues and eigenvectors? ›

      Mechanical Engineering: Eigenvalues and eigenvectors enable us to “decompose” a linear process into smaller, more manageable tasks. When stress is applied to a “plastic” solid, for example, the deformation can be divided into “principle directions,” or the directions where the deformation is greatest.

      How do you know if something is an eigenvalue? ›

      The scalar λ is an eigenvalue of A if Ax = λx for some nonzero vector x in Rn. If λ is an eigenvalue of A, then any vector x in Rn for which Ax = λx is an eigenvector of A corresponding to λ (or just λ-eigenvector).

      What are eigenvalues used for in finance? ›

      The eigenvalues and eigenvectors of a matrix are often used in the analysis of financial data and are integral in extracting useful information from the raw data. They can be used for predicting stock prices and analyzing correlations between various stocks, corresponding to different companies.

      What is the difference between eigen value and eigen function? ›

      Such an equation, where the operator, operating on a function, produces a constant times the function, is called an eigenvalue equation. The function is called an eigenfunction, and the resulting numerical value is called the eigenvalue. Eigen here is the German word meaning self or own.

      What is the difference between eigen value and eigen space? ›

      the eigenvalues are all the lambdas you find, the eigenvectors are all the v's you find that satisfy T(v)=lambda*v, and the eigenspace FOR ONE eigenvalue is the span of the eigenvectors cooresponding to that eigenvalue.

      What is the significance of eigenvalues and eigenvectors? ›

      They provide valuable information about the dynamics and flow of a system or problem. Eigenvalues represent the scalar values that determine the behavior of the system, while eigenvectors represent the corresponding vectors that define the directions of the system's behavior.

      Top Articles
      Latest Posts
      Article information

      Author: Gregorio Kreiger

      Last Updated:

      Views: 6499

      Rating: 4.7 / 5 (77 voted)

      Reviews: 92% of readers found this page helpful

      Author information

      Name: Gregorio Kreiger

      Birthday: 1994-12-18

      Address: 89212 Tracey Ramp, Sunside, MT 08453-0951

      Phone: +9014805370218

      Job: Customer Designer

      Hobby: Mountain biking, Orienteering, Hiking, Sewing, Backpacking, Mushroom hunting, Backpacking

      Introduction: My name is Gregorio Kreiger, I am a tender, brainy, enthusiastic, combative, agreeable, gentle, gentle person who loves writing and wants to share my knowledge and understanding with you.