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An eigenvector keeps its direction under a linear map; its eigenvalue gives the scale factor. The eigenvalues are found from the characteristic equation.
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Formula
A and λ are dimensionless when A is dimensionless · v ≠ 0
| Symbol | Meaning | Unit |
|---|---|---|
| Square matrix of the linear map | dependent on the units of A | |
| Nonzero eigenvector satisfying A·v = λ·v | context-dependent | |
| Eigenvalue, the scale factor in the direction of v | same unit as A | |
| Identity matrix with the same size as A | dimensionless |
\exists\,\mathbf{v}\ne\mathbf{0}: A\mathbf{v}=\lambda\mathbf{v} \quad\Longleftrightarrow\quad \det(A-\lambda I)=0An eigenvector is a direction that a linear map does not leave; it may be stretched, compressed or reversed. From A·v = λ·v we obtain (A − λI)·v = 0. For this homogeneous system to have a nontrivial solution v ≠ 0, A − λI must be singular, so det(A − λI) = 0. This equation gives the eigenvalues; for each value, the null space of A − λI then gives its eigenvectors. For real symmetric matrices, every eigenvalue is real and eigenvectors belonging to different eigenvalues are orthogonal.
A must be square; eigenvectors are nonzero by definition. Over ℂ, at least one eigenvalue exists. A real matrix can still have complex eigenvalues.
Nontrivial solutions exist exactly when A−λI is singular.
Form the characteristic equation
The determinant is a polynomial in λ; its roots are the eigenvalues.
Find an eigenvector for λ
All solutions of the homogeneous system form the eigenspace; the zero solution is not an eigenvector.
Find the eigenvalues and eigenvectors of A = [[2,0],[0,3]].
det(A−λI)=(2−λ)(3−λ)=0, so λ=2,3.
For λ=2, v=(1,0); for λ=3, v=(0,1).
Check: Av=λv.
Giving v = 0 as an eigenvector or solving the vector equation without the determinant condition.
Eigenvectors must be nonzero. First find λ from det(A−λI)=0, then solve the corresponding homogeneous system.
Formula cluster
Eigenvalues extend matrix methods and linear models with direction, scaling and dynamics.
For A = [[2,0],[0,3]], det(A − λI) = (2 − λ)(3 − λ) = 0. The eigenvalues are λ1 = 2 and λ2 = 3. For λ1, v1 = (1,0), since A·v1 = (2,0) = 2v1; for λ2, v2 = (0,1).
Linear dynamics and stability, vibration modes, principal component analysis, diagonalization and quantum-mechanical observables
Here is how to work through a typical Eigenvalues and Eigenvectors (A·v = λ·v · det(A − λI) = 0) task step by step:
Task
Find the eigenvalues and eigenvectors of A = [[2,0],[0,3]].
Solution path
det(A−λI)=(2−λ)(3−λ)=0, so λ=2,3.
For λ=2, v=(1,0); for λ=3, v=(0,1).
Check: Av=λv.
An eigenvector is a direction that a linear map does not leave. After multiplying v by A, Av remains parallel to v; only its length or orientation changes. The eigenvalue λ is the factor: λ = 2 doubles the length, λ = −1 reverses the direction, and λ = 0 maps the vector to zero. For A = diag(2,3), the x- and y-axes are scaled by different factors, so those axis directions are eigenvectors. Vectors between them usually are not. This geometric picture makes the algebra more understandable and helps you check whether a computed eigenvector is plausible.
First form A−λI and take its determinant. The characteristic equation det(A−λI)=0 has the eigenvalues as its roots. For A = [[2,0],[0,3]], the determinant is (2−λ)(3−λ), so the roots are 2 and 3. Then solve (A−λI)v=0 for each value to find its eigenvectors. Keep the order straight: the determinant equation gives λ, not v directly. Real matrices can have complex eigenvalues; for a real symmetric matrix, all eigenvalues are real. A repeated root can still have a one-dimensional eigenspace, so algebraic multiplicity does not count independent eigenvectors.
The eigenvector v is a nonzero direction; the eigenvalue λ is the number by which A scales that direction. They belong together and satisfy Av=λv. A matrix usually has several such pairs. For A = diag(2,3), v=(1,0) is an eigenvector with λ=2, while v=(0,1) has λ=3. The zero vector is explicitly not an eigenvector, even though A·0=λ·0 for every number. Multiples of the same v describe the same eigen-direction. This distinction prevents recording a result as only a number or only a vector.
A real symmetric matrix can be represented diagonally using an orthonormal basis of eigenvectors. The eigenvalues then sit on the diagonal and show the scaling along these perpendicular directions. This makes repeated matrix powers, quadratic forms and coupled oscillations easier to compute. Eigenvectors belonging to distinct eigenvalues are orthogonal; for a repeated eigenvalue, an orthonormal basis can be chosen within its eigenspace. This is more than a shortcut: in statistics it describes principal components, and in mechanics it describes normal modes. Not every matrix is symmetric, and complex or non-diagonalizable cases require extra care.
Eigenvalues show what happens to particular patterns or states when a model is applied repeatedly. In an oscillation problem they correspond to natural frequencies; in a dynamical system they reveal whether small disturbances decay or grow. In principal component analysis they give variance along new data directions. In quantum mechanics, measurable values are associated with eigenvalues of suitable operators. Each application has additional assumptions: stability, for example, depends on whether the system is continuous or discrete, and a data component is not a cause. The eigenvalue calculation supplies a structure; the model context determines its meaning.
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Which formula describes Eigenvalues and Eigenvectors?
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How do you rearrange Eigenvalues and Eigenvectors for Form the characteristic equation?
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Which common mistake happens with Eigenvalues and Eigenvectors?
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