ECON1202 Chap.9 Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
An eigenvector is a non-zero direction that a matrix transformation does not turn away from its line. Applying A produces a scalar multiple of the same vector, and the eigenvalue records the scale and possible reversal. Rearranging Av=lambda v gives a homogeneous system involving A-lambda I. A non-zero solution exists only when that matrix is singular, so the characteristic determinant must equal zero.
Solve this scalar equation first, then row-reduce A-lambda I for each eigenvalue and choose a non-zero vector from its solution family. Eigenvectors are not unique in length: every non-zero scalar multiple represents the same direction. For a two-by-two matrix, the eigenvalues sum to the trace and multiply to the determinant, providing two fast checks on the characteristic roots.
A triangular matrix reveals its eigenvalues on the diagonal. Direct verification compares Av with lambda v component by component. Repeated eigenvalues require care because root multiplicity does not by itself guarantee the same number of independent eigenvector directions.
What this chapter covers
- 01
Invariant directions
- 02
Eigenvalue scaling
- 03
Characteristic equation
- 04
Singularity of A minus lambda I
- 05
Two-by-two root checks
- 06
Triangular matrices
- 07
Eigenvector homogeneous systems
- 08
Direct eigenpair verification
Worked example · free
Find eigenpairs for a diagonal matrix
- +1Because A is triangular, its eigenvalues are the diagonal entries 4 and -2.
- +1For lambda=4, solve (A-4I)v=0 and choose v=[1,0]T.
- +1For lambda=-2, solve (A+2I)v=0 and choose v=[0,1]T.
- +1Verify A[1,0]T=4[1,0]T and A[0,1]T=-2[0,1]T.
Key terms
- Eigenvalue
- The scalar by which a matrix scales an associated eigenvector.
- Eigenvector
- A non-zero vector whose line is preserved by a matrix transformation.
- Characteristic equation
- The determinant equation det(A-lambda I)=0 used to find eigenvalues.
- Trace
- The sum of diagonal entries, equal to the sum of two-by-two eigenvalues.
- Invariant direction
- A line mapped into itself by the matrix transformation.
- Eigenpair
- An eigenvalue together with one associated non-zero eigenvector.
Eigenvalues and Eigenvectors FAQ
Why must an eigenvector be non-zero?
The zero vector satisfies Av=lambda v for every lambda and therefore identifies no special direction or scale. Excluding it makes the equation informative and requires A-lambda I to have a non-trivial homogeneous solution.
Why is the characteristic determinant set to zero?
A non-zero eigenvector solves (A-lambda I)v=0. Such a non-trivial solution exists only when A-lambda I is singular, and the determinant of a singular square matrix is zero.
How are eigenvalues checked for a two-by-two matrix?
Their sum should equal the trace and their product should equal the determinant. Use both after solving the characteristic equation; the checks confirm the roots but do not generate associated eigenvectors.
Can two different-looking eigenvectors both be correct?
Yes. Any non-zero scalar multiple of an eigenvector lies on the same invariant line and satisfies the same eigenvalue equation. Direct multiplication decides whether each proposed representative is valid.
What is the full reasoning chain for an eigenvalue question?
Begin from Av=lambda v and move all terms to obtain (A-lambda I)v=0. Explain that a non-zero solution exists only when this coefficient matrix is singular, which produces the characteristic equation det(A-lambda I)=0. Solve that scalar equation before finding directions.
For each root separately, row-reduce its homogeneous system, express dependent components in terms of free variables and choose a convenient non-zero representative. State that non-zero scalar multiples describe the same eigendirection. Verify every reported pair by calculating Av and lambda v side by side.
If an eigenvalue is repeated, do not assume the number of independent eigenvectors from multiplicity alone; solve the corresponding null space. Keep the zero vector outside the list of eigenvectors even though it belongs to every eigenspace as a subspace. Interpret the sign and magnitude of the eigenvalue only after the algebraic pair has been verified.
For a two-by-two matrix, the trace and determinant provide useful checks on the sum and product of the characteristic roots. Exact arithmetic should be retained through the verification so rounding does not disguise whether two vectors are truly proportional.
Exam move
Always separate the scalar stage from the vector stage. Form det(A-lambda I)=0, solve for eigenvalues, then handle each eigenvalue through its own homogeneous system. Use triangular examples to build intuition and general two-by-two examples to practise the characteristic polynomial. After finding roots, check their sum against the trace and product against the determinant.
When solving for vectors, mark a free variable and choose a convenient non-zero value; do not report the trivial vector. Verify every pair by calculating Av and lambda v side by side. Compare positive, negative, zero and fractional eigenvalues in words so scaling and reversal are understood geometrically.
For repeated roots, determine the solution space rather than assuming a number of independent directions from multiplicity alone.
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