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Applications of Eigenvalues and Eigenvectors

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Eigenvalue: The specific set of scalars connected with the system of linear equations is known as eigenvalues. Matrix equations are where it’s most commonly employed. The German term ‘Eigen’ denotes ‘appropriate’ or ‘characteristic.’ As a result, eigenvalue can also be referred to as a characteristic value, a characteristic root, appropriate values, or latent roots. To put it another way, an eigenvalue is a scalar that is utilized to convert an eigenvector. The fundamental formula is

Ax = λx

The eigenvalue of A is the number or scalar value “λ”.

Eigenvector: When a linear transformation is applied, eigenvectors are non-zero vectors that do not change direction. It only varies by scalar quantity. In a nutshell, if A is a linear transformation from a vector space V, and x is a non-zero vector in V, then v is an eigenvector of A if A(X) is a scalar multiple of x. A set of all the eigenvectors with the identical eigenvalue, jointly with the zero vector, makes up an Eigenspace of vector x. The zero vector, however, is not an eigenvector.  If A is an “n n” matrix and is an eigenvalue of A, then x, a non-zero vector, is called an eigenvector if it fulfills the following expression:

Ax = λx

x is one of the eigenvectors of A value that corresponds to eigenvalue λ.

Applications of a Square Matrix’s Eigenvalues and Eigenvectors

1. System of Communication: Claude Shannon utilized eigenvalues to calculate the theoretical limit of how much information can be carried via a communication channel such as a telephone line or the air. The eigenvectors and eigenvalues of the communication channel (represented as a matrix) are calculated, and then the eigenvalues are waterfilled. The eigenvalues are then essentially the gains of the channel’s fundamental modes, which are recorded by the eigenvectors.

2. Bridge Construction: The smallest magnitude eigenvalue of a system that models the bridge is the natural frequency of the bridge. Engineers use this knowledge to guarantee that their structures are stable.

3. Automobile Stereo System Design: Eigenvalue analysis is also employed in the design of car stereo systems, where it aids in the reproduction of car vibration caused by music.

4. Electrical Engineering: The use of eigenvalues and eigenvectors to decouple three-phase systems via symmetrical component transformation is advantageous.

5. 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. The eigenvectors in the principle directions are the eigenvectors, and the associated eigenvalue is the percentage deformation in each principle direction.

Eigenvalue analysis is commonly used by oil firms to explore land for oil. Because oil, dirt, and other substances all produce linear systems with varying eigenvalues, eigenvalue analysis can help pinpoint where oil reserves lie. Oil companies set up probes all-around a site to pick up the waves created by a massive truck vibrating the ground. The waves are modified when they move through the different substances in the earth. The oil corporations are directed to possible drilling sites based on the study of these waves.


Last Updated : 17 Feb, 2022
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