Range and Null Space
Eigenvectors are non-zero vectors that a matrix transforms only by scaling.
The Exceptional Direction
A matrix usually changes a vector in more than one way: its length may change, and its direction may change as well. An eigenvector is an exceptional direction. When the matrix acts on an eigenvector, the result remains on the same line as the original vector. The eigenvalue records the scalar scaling associated with that direction.
Testing the Eigenvector Relation
A non-zero vector u is an eigenvector of a matrix A when the matrix transforms it only by scaling. The defining relationship is Au = λu, where λ is the eigenvalue associated with u.
The equation Au = λu is a complete test. First calculate Au. Then ask whether the result can be written as one scalar multiplied by u. If it can, u is an eigenvector and that scalar is its eigenvalue. If the output points in a different direction instead of remaining a scalar multiple of u, the candidate is not an eigenvector for A.
Recognizing a Scaling Relationship
Suppose u is a non-zero vector and a matrix A produces Au = 3u. What does this tell us?
Check the form: The result Au is written as a scalar, 3, multiplied by the original vector u.
Identify the eigenvalue: The scalar multiplying u is the eigenvalue, so λ is 3.
Identify the eigenvector: Because u is specified to be non-zero and A transforms it only by scaling, u is an eigenvector of A.
u is an eigenvector of A, and its associated eigenvalue is 3.
Why Zero Is Excluded
An eigenvector must be non-zero. If the zero vector were allowed, then A times the zero vector would equal the zero vector for every matrix and every scalar. In that case, every scalar could appear to be an eigenvalue, so the zero vector would not identify a meaningful preserved direction.
| Candidate | What happens under A | Why it matters |
|---|---|---|
| Non-zero vector u | Au may equal λu for one associated scalar λ | It can identify an eigenvector direction and its scaling. |
| Zero vector | A times the zero vector equals the zero vector | Every scalar would satisfy the equation, so no specific eigenvalue or direction is identified. |
Symmetric Matrix Decomposition
For a symmetric matrix A of rank k, the spectral decomposition theorem organizes its eigenvectors into an orthonormal basis of the surrounding vector space. If the basis vectors are u1 through ud, they are mutually perpendicular unit directions. The matrix U has these eigenvectors as its columns. The matrix D is diagonal: its ith diagonal entry is the eigenvalue λi associated with ui, and its off-diagonal entries are zero.
A = UDUᵀThis factorization says that A can be reconstructed from two kinds of information: the orthonormal eigenvector directions stored in U and the corresponding scaling values stored on the diagonal of D. It is therefore more than a formula; it gives a structural view of how the matrix acts along its important directions.
Nonzero Directions and Range
The eigenvalues separate the important directions of a symmetric matrix. An eigenvector paired with a nonzero eigenvalue contributes a direction to the range of A. The eigenvectors paired with nonzero eigenvalues span the range of A. The number of nonzero eigenvalues equals the rank of A.
Reading Rank and Range from Eigenvalues
A symmetric matrix has eigenvalues λ₁, λ₂, and λ₃, where λ₁ and λ₂ are nonzero and λ₃ is zero. What structural conclusions follow?
Count nonzero eigenvalues: There are two nonzero eigenvalues.
Identify the rank: The number of nonzero eigenvalues equals the rank, so the rank is 2.
Identify the range directions: The eigenvectors paired with λ₁ and λ₂ span the range of A.
Separate the zero direction: The eigenvector paired with λ₃ belongs to the null space because λ₃ is zero.
The matrix has rank 2; the eigenvectors for λ₁ and λ₂ span the range, while the eigenvector for λ₃ spans a null-space direction.
Zero Directions and Null Space
When an eigenvalue is zero, the defining relationship becomes Au = 0u, which is the zero vector. Therefore, an eigenvector paired with a zero eigenvalue is a vector in the null space of A. The eigenvectors paired with zero eigenvalues span the null space.
The zero eigenvalue does not describe a direction that is merely scaled by a positive or negative amount. It identifies a direction that the matrix sends to zero. Collecting all eigenvectors associated with zero eigenvalues gives the null-space directions in the symmetric-matrix decomposition.
Mistakes to Avoid
Treating any vector that satisfies Au = λu as an eigenvector without checking whether it is non-zero.
The zero vector would satisfy the equation for every scalar, so it would not identify a specific eigenvalue or direction.
Fix:
Require the candidate vector to be non-zero before applying the eigenvector test.Calling a vector an eigenvector when A changes its direction.
An eigenvector must remain on the same line after the matrix acts on it.
Fix:
Calculate Au and check whether it is a scalar multiple of the original vector.Associating the range with the zero-eigenvalue directions.
Zero-eigenvalue eigenvectors span the null space, while nonzero-eigenvalue eigenvectors span the range.
Fix:
Use nonzero eigenvalues to identify range directions and zero eigenvalues to identify null-space directions.Reading U as a diagonal matrix of eigenvalues.
U contains eigenvectors as columns, while D contains eigenvalues on its diagonal.
Fix:
Keep the roles separate: U stores orthonormal directions and D stores their corresponding scalings.
Check Your Understanding
A non-zero vector u satisfies Au = 0u. Identify the eigenvalue and state which space contains u. Then explain what changes if another eigenvector has a nonzero eigenvalue.
Hints
- Read the scalar multiplying u in Au = λu.
- Use the distinction between zero-eigenvalue and nonzero-eigenvalue directions.
- Remember which eigenvectors span the null space and which span the range.
- An eigenvector is a non-zero vector u satisfying Au = λu. The eigenvalue λ records the scaling while the vector's direction is preserved. For a symmetric matrix, A = UDUᵀ organizes orthonormal eigenvector directions in U and their eigenvalues in the diagonal matrix D. Nonzero eigenvalues identify eigenvectors that span the range, and the number of nonzero eigenvalues equals the rank. Zero eigenvalues send their eigenvectors to the zero vector, so those eigenvectors span the null space.
Key Takeaways
- Eigenvectors are non-zero vectors whose direction is preserved by a matrix, so Au = λu.
- The zero vector is excluded because every scalar would satisfy the eigenvector equation for it.
- For a symmetric matrix, A = UDUᵀ uses orthonormal eigenvectors in U and corresponding eigenvalues in D.
- The number of nonzero eigenvalues equals the rank, and their eigenvectors span the range.
- Eigenvectors associated with zero eigenvalues span the null space.