Singular Value Decomposition
A symmetric matrix is positive definite when every eigenvalue is positive.
Why the Zero Matters
Two matrix classifications can look almost identical: positive definite and positive semidefinite. Both are decided by examining eigenvalues of a symmetric matrix. The decisive difference is whether zero is allowed. If every eigenvalue is positive, the matrix is positive definite. If every eigenvalue is merely nonnegative, the matrix is positive semidefinite, so zero eigenvalues are permitted.
When classifying a matrix under the supplied definition, check two things in order: the matrix must be symmetric, and then its eigenvalues must satisfy the appropriate sign condition.
The Eigenvalue Test
The supplied definition places symmetry at the beginning of the classification. For a symmetric matrix, inspect every eigenvalue. Positive definiteness requires every eigenvalue to be greater than zero. Positive semidefiniteness requires every eigenvalue to be greater than or equal to zero. Therefore, a zero eigenvalue rules out positive definiteness but does not rule out positive semidefiniteness.
| Classification | Matrix condition in the supplied definition | Allowed eigenvalues | Role of zero |
|---|---|---|---|
| Positive definite | The matrix is symmetric and every eigenvalue is positive | Only positive eigenvalues | Not allowed |
| Positive semidefinite | The matrix is symmetric and every eigenvalue is nonnegative | Positive or zero eigenvalues | Allowed |
A Classification Walkthrough
Three eigenvalue sets
Assume each matrix is symmetric. Classify the eigenvalue sets {1, 4, 6}, {0, 0, 5}, and {-2, 0, 3}.
Set {1, 4, 6}: Every eigenvalue is positive. Because the matrix is assumed to be symmetric, this satisfies the supplied definition of positive definite.
Set {0, 0, 5}: Every eigenvalue is nonnegative, so the set satisfies the supplied definition of positive semidefinite. It is not positive definite because zero eigenvalues are present.
Set {-2, 0, 3}: The negative eigenvalue prevents every eigenvalue from being positive and also prevents every eigenvalue from being nonnegative. Under the supplied classifications, the matrix is neither positive definite nor positive semidefinite.
The first set is positive definite, the second is positive semidefinite but not positive definite, and the third is neither.
The example shows why the word symmetric cannot be treated as decoration. In the supplied definition, symmetry establishes the setting in which the eigenvalue test is being stated. After that condition is in place, the signs of the eigenvalues determine the classification.
What the Source Says About SVD
Singular Value Decomposition, abbreviated SVD, is introduced in the supplied material as a decomposition technique. The material identifies SVD as a matrix decomposition topic, but it does not provide a decomposition formula, name component matrices, or describe individual computational steps.
Common Classification Errors
Treating nonnegative as if it meant positive
Positive definiteness requires every eigenvalue to be positive. The zero eigenvalue violates that requirement.
Fix:
Classify this matrix as positive semidefinite, because every eigenvalue is nonnegative and zero is allowed in that case.Ignoring symmetry
The supplied definition begins with a symmetric matrix. Symmetry is part of the definition being taught.
Fix:
Keep the scope explicit: first recognize the symmetry requirement, then apply the eigenvalue sign test.Calling a matrix with a negative eigenvalue positive semidefinite
Positive semidefiniteness requires every eigenvalue to be nonnegative, not merely some of them.
Fix:
A negative eigenvalue means the set satisfies neither supplied classification.Adding unsupported SVD details
The supplied material introduces SVD as a decomposition technique but does not specify its formula, components, or steps.
Fix:
State only the source-supported introduction unless a later section supplies more detail.
Practice Check
What do you think happens?
Assume a matrix is symmetric and has eigenvalues 0, 2, and 8. Which classification applies?
Reveal answer
Answer: Positive semidefinite
All eigenvalues are nonnegative, so the matrix meets the supplied positive semidefinite condition. The zero eigenvalue prevents it from being positive definite.
Assume each matrix is symmetric. Classify these eigenvalue sets as positive definite, positive semidefinite, or neither: {3, 5, 9}; {0, 4, 7}; and {-1, 2, 6}. Then describe, in one sentence, what the supplied material introduces SVD as.
Hints
- Positive definite requires every eigenvalue to be positive.
- Positive semidefinite requires every eigenvalue to be nonnegative.
- A zero eigenvalue is allowed for positive semidefinite but not positive definite.
- A negative eigenvalue prevents both supplied classifications.
- The source introduces SVD as a decomposition technique.
Key Takeaways
- Under the supplied definition, a symmetric matrix is positive definite when every eigenvalue is positive.
- A symmetric matrix is positive semidefinite when every eigenvalue is nonnegative.
- Zero eigenvalues are excluded from positive definiteness but allowed in positive semidefiniteness.
- Symmetry is part of the supplied definition, so it belongs before the eigenvalue sign test in the classification process.
- The supplied material introduces Singular Value Decomposition, or SVD, as a matrix decomposition technique without specifying its formula, components, or computational steps.
Key Takeaways
- For a symmetric matrix, positive definiteness means that every eigenvalue is positive.
- Positive semidefiniteness means that every eigenvalue is nonnegative.
- The presence of a zero eigenvalue separates the two classifications: it is allowed for positive semidefinite matrices but not for positive definite matrices.
- A negative eigenvalue prevents a symmetric matrix from meeting either supplied classification.
- SVD is introduced as a matrix decomposition technique, but the supplied material does not give its formula, components, or computational steps.