Positive Semidefinite Matrices
A symmetric matrix is positive definite when every eigenvalue is positive.
One Zero Changes the Classification
The central classification rule is short: first consider a symmetric matrix, then inspect its eigenvalues. If every eigenvalue is positive, the matrix is positive definite. If every eigenvalue is nonnegative, the matrix is positive semidefinite. The difference is whether zero is allowed.
Start with Symmetry
Symmetry is not an optional detail in the supplied definition. The matrices being classified are described as symmetric. A matrix is symmetric when its entries correspond across the main diagonal: the entry in one position has the same value as the entry in the reflected position across that diagonal. Once this structural condition is in place, the eigenvalue signs provide the classification test.
The supplied rule is not simply an eigenvalue rule for every possible matrix. It specifically describes a symmetric matrix as positive definite or positive semidefinite, depending on the signs of its eigenvalues.
Trace the Eigenvalue Test
After symmetry has been identified, examine every eigenvalue rather than checking only one. The word every is decisive. A symmetric matrix is positive definite only when no eigenvalue is zero and none is negative. A symmetric matrix is positive semidefinite when each eigenvalue is either positive or zero. A single negative eigenvalue prevents the matrix from satisfying either of these supplied classifications.
| Eigenvalue condition for a symmetric matrix | Classification |
|---|---|
| Every eigenvalue is positive | Positive definite |
| Every eigenvalue is nonnegative; zero is allowed | Positive semidefinite |
| At least one eigenvalue is negative | Neither classification under the supplied rule |
Classification depends on the sign of every eigenvalue.
Classify Three Eigenvalue Lists
Reading the Signs
Assume each listed set of eigenvalues belongs to a symmetric matrix. Classify each set using the supplied definition.
List A: The eigenvalues are 2, 5, and 9. Every value is positive, so the matrix is positive definite.
List B: The eigenvalues are 0, 3, and 7. Every value is nonnegative, and zero is present. The matrix is positive semidefinite, not positive definite.
List C: The eigenvalues are -1, 2, and 4. A negative eigenvalue is present, so the matrix is neither positive definite nor positive semidefinite under the supplied rule.
The decisive scan is: all positive means positive definite; positive values together with possible zeros mean positive semidefinite; any negative value rules out both classifications.
What do you think happens?
A symmetric matrix has eigenvalues 0, 0, and 6. Which classification fits the supplied definitions?
Reveal answer
Answer: Positive semidefinite
All eigenvalues are nonnegative, so the matrix satisfies the positive semidefinite condition. The zero eigenvalues prevent it from being positive definite.
Avoid the Zero-Eigenvalue Trap
Treating positive semidefinite as a synonym for positive definite.
The zero eigenvalue is permitted for the semidefinite classification but excluded from the definite classification.
Fix:
Classify it as positive semidefinite, not positive definite.Checking only whether some eigenvalues are positive.
The supplied tests require every eigenvalue to satisfy the relevant sign condition.
Fix:
Check the complete list. The negative eigenvalue rules out both classifications.Ignoring symmetry and applying the supplied classification rule without mentioning it.
The supplied definition begins with a symmetric matrix.
Fix:
Recognize the symmetry condition as part of the definition, then inspect the eigenvalues.
Use a two-stage checklist: first record that the matrix is symmetric, then inspect every eigenvalue. This prevents the most common classification error: overlooking a zero or a negative value.
Locate SVD in the Topic
The supplied material also introduces Singular Value Decomposition, abbreviated SVD, as a matrix decomposition technique. In this source section, SVD is introduced as a topic alongside the review of positive definite and positive semidefinite matrices. The supplied material does not provide an SVD formula, name its component matrices, or describe individual computational steps. Therefore, the source-supported takeaway is limited to its role as a decomposition topic introduced in the same section.
Practice the Classification
For each case, assume the matrix is symmetric and classify it as positive definite, positive semidefinite, or neither: (1) eigenvalues 1, 4, and 12; (2) eigenvalues 0, 2, and 10; (3) eigenvalues -3, 1, and 6. Then explain in one sentence why zero changes the result in the second case.
Hints
- Positive definite requires every eigenvalue to be positive.
- Positive semidefinite permits zero but requires every eigenvalue to be nonnegative.
- A single negative eigenvalue is enough to rule out both classifications under the supplied rule.
A reliable answer names both parts of the test: the matrix is treated as symmetric, and the complete eigenvalue list satisfies either the strictly positive condition or the nonnegative condition.
Keep the Decision Rule
- The supplied definitions apply to symmetric matrices.
- Every positive eigenvalue gives the positive definite classification.
- Positive semidefiniteness allows zero eigenvalues as long as every eigenvalue is nonnegative.
- A negative eigenvalue prevents either supplied classification.
- SVD is introduced in the source as a matrix decomposition technique, without a supplied formula, component list, or computational procedure.
Key Takeaways
- For a symmetric matrix, inspect the sign of every eigenvalue.
- Every positive eigenvalue means positive definite.
- Every nonnegative eigenvalue means positive semidefinite, so zero is allowed.
- Any negative eigenvalue rules out both classifications under the supplied definition.
- The supplied section introduces SVD as a decomposition technique but does not specify its formula or components.