Eigenvalues

Precise definition

A scalar $\lambda$ is an eigenvalue of a square matrix $A$ if there exists a nonzero vector $v$ with $Av=\lambda v$. Equivalently, $A-\lambda I$ is singular, so eigenvalues are roots of the characteristic equation $\det(A-\lambda I)=0$.

Notation and mathematical language

The characteristic polynomial has degree $n$ for an $n\times n$ matrix over any field. Over an algebraic closure, it has $n$ roots counting algebraic multiplicity. Those roots sum to $\operatorname{tr}A$ and multiply to $\det A$.

Conceptual picture

An eigenvalue is a scaling factor along an invariant direction. It can be negative, reversing direction, or complex, encoding rotation-scaling in a complexified space. Eigenvalues depend on the linear transformation, not the chosen basis: similar matrices share them.

Conditions and key results

Only square matrices have eigenvalues in this standard sense. Solving the characteristic polynomial identifies possible scalars, but an eigenvector must be nonzero. Real matrices can have nonreal eigenvalues; repeated roots may have smaller eigenspaces.

A reliable strategy

1. Confirm the matrix is square and form $A-\lambda I$ with signs consistent.
2. Compute and factor the characteristic polynomial over the requested field.
3. Record algebraic multiplicities and use trace and determinant as checks.
4. For interpretation or diagonalization, solve each eigenspace and distinguish eigenvalues from singular values.

Fully worked example

Interpretation and application

Eigenvalues determine modes in differential equations, long-term matrix powers, stability, and principal directions for symmetric matrices. A large eigenvalue magnitude need not mean a general nonnormal matrix stretches every vector that much in one step.

Common mistakes

Verification and reasonableness

• Check $Av=\lambda v$ for a sample vector from each eigenspace.
• Compare eigenvalue sum/product with trace/determinant.
• Substitute roots back into the characteristic polynomial.

Practice

1. Find eigenvalues of $\operatorname{diag}(2,-3)$.
2. Is zero an eigenvalue exactly when $A$ is singular?
3. Can a real matrix have complex eigenvalues?

Further deduction

The spectral radius $\rho(A)=\max|\lambda_i|$ governs asymptotic powers in many finite-dimensional settings: if $A$ is diagonalizable, each eigenmode gains a factor $\lambda_i^k$. Jordan blocks add polynomial factors in $k$, but when $\rho(A)<1$, all powers still tend to zero. This conclusion concerns repeated iteration, not necessarily monotone norm decrease at every step.

Gershgorin's theorem gives a useful bound: every eigenvalue lies in at least one disk centred at a diagonal entry $a_{ii}$ with radius $\sum_{j\ne i}|a_{ij}|$. It can catch arithmetic errors in computed roots and sometimes proves positivity or stability regions, although it generally bounds rather than determines exact eigenvalues.

The characteristic polynomial supplies fast consistency checks: over the complex numbers and counting algebraic multiplicity, the sum of eigenvalues equals $\operatorname{tr}A$ and their product equals $\det A$. For a triangular matrix the eigenvalues are its diagonal entries, immediately matching both identities. These checks can expose a lost sign or root but do not by themselves produce eigenvectors or guarantee diagonalizability; multiplicity and eigenspace dimension still have to be examined.

Related topics

• Eigenvectors
• Diagonalization
• Determinants