- Explain eigenvalues and eigenvectors through the equation A x = λx.
- Find eigenvalues from a characteristic equation and determine their eigenvectors.
- Check eigenvalues using the trace and determinant.
- Distinguish algebraic multiplicity from geometric multiplicity.
- Relate distinct or repeated eigenvalues to linear independence and diagonalizability.
- Matrix multiplication
- Determinants of square matrices
- Homogeneous linear systems
- Rank and linear independence
The central idea
An eigenvector keeps its line of action under a square-matrix transformation, while its eigenvalue records the scale factor.
Think of a square matrix A as a transformation that may stretch, rotate, or compress vectors. Most vectors change direction, but certain nonzero vectors retain the same direction or reverse it. Such a vector is an eigenvector, and its scale factor λ is the corresponding eigenvalue.
Move both terms to one side and factor out the vector to obtain a homogeneous system.
Here I is the identity matrix with the same dimensions as A. A homogeneous system is always consistent because x = 0 is a solution. Eigenvectors must be nonzero, however, so the system must admit infinitely many solutions. This occurs precisely when A − λI is singular and therefore has determinant zero.
Build and solve the characteristic equation
The determinant condition produces a polynomial whose roots are the eigenvalues.
The expression det(A − λI) is the characteristic polynomial. Setting it equal to zero forms the characteristic equation, and its roots give every eigenvalue.
Why does the method use a determinant, and what prevents its direct use for a nonsquare matrix?
A nonzero solution requires A − λI to be singular, which is detected by a zero determinant. The determinant and this eigenvalue construction apply to square matrices.
Solve for the eigenvectors
Insert each eigenvalue into the homogeneous system and describe its nonzero solutions.
After finding λ, substitute it into (A − λI)x = 0. Because the determinant is zero, the shifted matrix has reduced rank. In a 2×2 problem the rank may fall to 1, or even 0, leaving free variables and infinitely many solutions. Every nonzero solution is an eigenvector for that λ.
Verify with trace and determinant
The trace checks the eigenvalue sum, while the determinant checks their product.
For every square matrix, the trace equals the sum of its eigenvalues and the determinant equals their product, with multiplicities included.
- trace(A) = sum of the eigenvalues, counting repeated roots.
- det(A) = product of the eigenvalues, counting repeated roots.
Algebraic and geometric multiplicity
Repeated roots are measured in the characteristic polynomial and in their eigenspaces.
For (λ − 1)²(λ − 3), the eigenvalue 1 has AM = 2, whereas 3 has AM = 1.
A matrix is diagonalizable when AM and GM agree for every eigenvalue, allowing a basis of eigenvectors. If some eigenvalue has GM smaller than AM, the matrix is defective and cannot be diagonalized.
When eigenvectors are independent
Distinct eigenvalues guarantee independent eigenvectors; repeated roots depend on geometric multiplicity.
- Eigenvectors belonging to distinct eigenvalues are always linearly independent.
- If the eigenvalues are 1, 2, and 3, one corresponding eigenvector for each value gives three linearly independent vectors.
- With repeated eigenvalues, the associated eigenvectors may or may not supply enough independent directions.
- Geometric multiplicity gives the exact number of independent eigenvectors for that eigenvalue.
When GM reaches AM, all required eigenvector directions are present. When it does not, some directions are missing and diagonalization is impossible.
Complete example from start to finish
Apply the full method, verify the roots, and interpret the result.
For A = ((6, −2), (2, 1)), what eigenvector families correspond to λ = 2 and λ = 5?
For λ = 2, x = ᵀ. For λ = 5, x = ᵀ, with k nonzero in both cases.
Review the full procedure
Follow the same sequence for any square-matrix eigenvalue problem.
- Construct A − λI by subtracting λ from every diagonal entry.
- Evaluate det(A − λI) to produce the characteristic polynomial.
- Solve det(A − λI) = 0 to obtain all eigenvalues.
- For each λ, solve (A − λI)x = 0 and keep the nonzero solutions.
- Compare the eigenvalue sum with the trace and their product with the determinant.
- Read AM from the factor's exponent and GM from the number of independent eigenvectors.
- Use the multiplicities and whether the roots are distinct to assess eigenvector independence and diagonalizability.
Which equation must be solved to find a square matrix's eigenvalues?
Solve det(A − λI) = 0.
How are algebraic and geometric multiplicity related?
For every eigenvalue, AM ≥ GM.
Common misconceptions
Watch out: The zero vector can be counted as an eigenvector because it solves the homogeneous system.
Instead: Although x = 0 always solves the system, an eigenvector must be nonzero.
Watch out: A repeated eigenvalue automatically supplies as many independent eigenvectors as its algebraic multiplicity.
Instead: Its geometric multiplicity may be smaller than its algebraic multiplicity; equality must be checked.
Watch out: The spectral radius is always the numerically largest eigenvalue.
Instead: It is the largest eigenvalue in absolute value.
Lesson recap
- Eigenpairs satisfy .
- Eigenvalues are the roots of det(A − λI) = 0.
- Eigenvectors are the nonzero solutions of (A − λI)x = 0 for each root.
- The eigenvalue sum equals the trace, and their product equals the determinant.
- Algebraic multiplicity counts repeated roots; geometric multiplicity counts independent eigenvector directions, with AM ≥ GM.
- The spectrum contains all eigenvalues, and the spectral radius is the largest absolute eigenvalue.