Matrix Rank
Eigenvectors are non-zero vectors that a matrix transforms only by scaling.
A Direction That Survives Transformation
A matrix usually changes a vector in more than one way. Its output may have a different length and a different direction. An eigenvector is an exceptional vector: after the matrix acts on it, the result remains on the same line as the original vector. The eigenvalue records the scaling associated with that direction.
Testing the Eigenvector Relationship
A non-zero vector u is an eigenvector of a matrix A when applying A produces a scalar multiple of u. The relationship is Au = λu. Here, λ is the eigenvalue associated with u.
- Start with a candidate vector u.
- Calculate the transformed vector Au.
- Ask whether Au can be written as one scalar times u.
- If the answer is yes, that scalar is the eigenvalue λ and u is an eigenvector.
- If the output points in a different direction instead of remaining a scalar multiple of u, the candidate is not an eigenvector for A.
A Small Transformation Trace
Separating Preserved Directions
Consider the symmetric matrix A that scales one coordinate direction by 3 and sends a second coordinate direction to zero. Use the eigenvector relationship to identify the two associated eigenvalues and interpret the directions.
First direction: Let u1 be the first coordinate direction. By construction in this generated example, A transforms u1 into 3u1. Therefore u1 satisfies Au1 = 3u1, so its eigenvalue is 3.
Second direction: Let u2 be the second coordinate direction. A transforms u2 into the zero vector, which is 0u2. Therefore u2 satisfies Au2 = 0u2, so its eigenvalue is zero.
Interpretation: The first direction survives as a nonzero scaled direction. The second direction is eliminated by the transformation and belongs to the null-space directions.
The two directions have eigenvalues 3 and 0. The nonzero-eigenvalue direction contributes to the range, while the zero-eigenvalue direction belongs to the null space.
Organizing a Symmetric Matrix
For a symmetric matrix A, the spectral decomposition theorem organizes the matrix through an orthonormal basis of eigenvectors. If the surrounding vector space has basis vectors u1 through ud, place these eigenvectors as the columns of U. The matrix D is diagonal: its ith diagonal entry is the eigenvalue λi associated with ui, and its off-diagonal entries are zero.
A = UDUᵀThe factorization gives a structural description of the transformation. U supplies mutually perpendicular unit directions, while D records how the matrix scales each corresponding direction. Reconstructing A as UDUᵀ combines those directions and scaling values into the original symmetric matrix.
Reading Rank from Eigenvalues
For the symmetric-matrix setting described by the spectral decomposition, the number of nonzero eigenvalues equals the rank of the matrix. Each eigenvector paired with a nonzero eigenvalue identifies a direction that contributes to the matrix's range. Those eigenvectors span the range of A.
| Eigenvalue type | Associated direction | Structural role |
|---|---|---|
| Nonzero | Eigenvector paired with a nonzero scaling | Contributes to the range and to the rank |
| Zero | Eigenvector sent to the zero vector | Belongs to the null space |
The eigenvalue separates directions that remain in the range from directions that are eliminated.
Tracing the Null-Space Directions
An eigenvector paired with a zero eigenvalue satisfies Au = 0u, so the matrix sends that eigenvector to the zero vector. Such eigenvectors span the null space of A. In the decomposition, these are the directions whose scaling value is zero rather than nonzero.
When interpreting a spectral decomposition, sort the eigenvector directions by their eigenvalues conceptually: nonzero eigenvalues identify range directions, and zero eigenvalues identify null-space directions. This keeps the geometric and rank interpretations connected to the diagonal entries of D.
Common Interpretation Errors
Treating the zero vector as an eigenvector.
This equality is automatic and does not identify a meaningful direction or a specific eigenvalue.
Fix:
Use the definition only with a non-zero vector.Calling every transformed vector an eigenvector.
An eigenvector must produce a scalar multiple of itself, not merely some output vector.
Fix:
Check whether Au can be written as λu for one scalar λ.Counting all eigenvalues as rank contributions.
The source relationship assigns rank contributions to nonzero eigenvalues; zero eigenvalues identify null-space directions.
Fix:
Count the nonzero eigenvalues when interpreting rank in the symmetric-matrix setting.Describing U as a collection of arbitrary vectors.
In the spectral decomposition of a symmetric matrix, the columns of U are an orthonormal eigenvector basis.
Fix:
Remember that U contains mutually perpendicular unit eigenvector directions.
Practice and Final Check
A symmetric matrix has an orthonormal eigenvector basis. Two eigenvectors have nonzero eigenvalues, and the remaining eigenvectors have eigenvalue zero. Explain what the two groups tell you about the matrix's rank, range, and null space.
Hints
- Use the number of nonzero eigenvalues to identify the rank.
- The eigenvectors paired with nonzero eigenvalues span the range.
- The eigenvectors paired with zero eigenvalues span the null space.
- The central test for an eigenvector is Au = λu, with u required to be non-zero. For a symmetric matrix, the spectral decomposition A = UDUᵀ uses an orthonormal eigenvector basis in U and the corresponding eigenvalues on the diagonal of D. Nonzero eigenvalues identify eigenvector directions that span the range and determine the rank. Zero eigenvalues identify eigenvector directions that are sent to zero and span the null space.
Key Takeaways
- An eigenvector is a non-zero vector that a matrix transforms into a scalar multiple of itself.
- The eigenvalue is the scalar in Au = λu and records the scaling of the eigenvector direction.
- For a symmetric matrix, A = UDUᵀ organizes an orthonormal eigenvector basis and its associated eigenvalues.
- Nonzero eigenvalues correspond to directions spanning the range and their count equals the matrix rank.
- Zero eigenvalues correspond to directions spanning the null space.