Finding Eigenvalues and Eigenvectors | 3×3 Matrix 2×2 Matrix Example | Linear Algebra

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Matrix acts by stretching the vector , not changing its direction, so is an eigenvector of . where, for each index , . then v is an eigenvector of the linear transformation A and the scale factor λ is the eigenvalue corresponding to that eigenvector.
In linear algebra, an eigenvector or characteristic vector of a square matrix is a vector that points in a direction which is invariant under the associated linear transformation. In other words, if v is a vector which is not zero, then it is an eigenvector of a square matrix A if Av is a scalar multiple of v. where λ is a number (also called a scalar) known as the eigenvalue or characteristic value associated with the eigenvector v.
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Real matrices
Two dimensional example
Three dimensional example
Diagonal matrices
Triangular matrices
Eigenvector basis
Matrices
Characteristic polynomial
Diagonalization and eigendecomposition
Left and right eigenvectors
Variational characterization
Geometric multiplicity
Eigenspace and spectrum
Eigenbasis
Dynamic equations
Calculation
Eigenvalues
Eigenvectors
Generalizations to infinite-dimensional spaces
Eigenfunctions
Spectral theory
Associative algebras and representation theory
Applications
Eigenvalues of geometric transformations
Schrödinger equation
Molecular orbitals
Geology and glaciology
Principal components analysis
Vibration analysis
Eigenfaces
Tensor of moment of inertia
Stress tensor
Graphs
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