Symmetric Matrices
A symmetric matrix is positive definite when every eigenvalue is positive.
The Classification Question
When classifying a symmetric matrix in this section, focus on one property of its eigenvalues: their signs. The decisive boundary is whether every eigenvalue is strictly positive or whether zero is also allowed.
Eigenvalue Sign Test
The supplied definition uses a two-level test. A symmetric matrix is positive definite when every eigenvalue is positive. It is positive semidefinite when every eigenvalue is nonnegative. The word nonnegative includes zero, so the semidefinite category is less strict at that boundary.
Symmetry in Scope
In the supplied material, the eigenvalue definitions are stated for a symmetric matrix. Symmetry therefore identifies the class of matrices to which this section's positive definite and positive semidefinite descriptions apply.
Classification Walkthrough
Two Symmetric-Matrix Classifications
Classify two hypothetical symmetric matrices from their eigenvalue lists.
- 1Case A
Suppose the eigenvalues are 2, 1, and 4. Every eigenvalue is positive, so the matrix satisfies the supplied definition of positive definite.
- 2Case B
Suppose the eigenvalues are 3, 0, and 5. Every eigenvalue is nonnegative, so the matrix is positive semidefinite. The zero eigenvalue prevents it from being positive definite.
- 3Compare
Both cases have no negative eigenvalues, but only Case A has strictly positive eigenvalues throughout.
Case A is positive definite. Case B is positive semidefinite but not positive definite.
The walkthrough isolates the only difference needed for these two labels: positive definiteness requires strict positivity for every eigenvalue, while positive semidefiniteness accepts the boundary value zero.
SVD Introduction
The source section also introduces Singular Value Decomposition, abbreviated SVD, as a matrix decomposition technique. In the supplied material, SVD appears as a topic alongside the review of positive definite and positive semidefinite matrices.
Frequent Classification Errors
Common mistake Treating positive semidefinite as requiring strictly positive eigenvalues.
What it looks likeA symmetric matrix with eigenvalues 3, 0, and 5 is rejected as positive semidefinite because one eigenvalue is zero.
Why it failsThe semidefinite definition allows every eigenvalue to be nonnegative, and zero is nonnegative.
Do this insteadClassify this eigenvalue list as positive semidefinite. It is not positive definite because positive definiteness excludes zero.
Common mistake Treating a zero eigenvalue as compatible with positive definiteness.
What it looks likeA learner calls a matrix with eigenvalues 2, 0, and 4 positive definite.
Why it failsPositive definiteness requires every eigenvalue to be positive.
Do this insteadUse the positive semidefinite classification when all eigenvalues are nonnegative and at least one is zero.
Common mistake Ignoring the symmetry condition in the supplied definition.
What it looks likeA learner applies the section's definition without checking that the matrix belongs to the stated symmetric-matrix setting.
Why it failsThe supplied definition is phrased for a symmetric matrix.
Do this insteadState the symmetry setting before applying the eigenvalue sign test.
Common mistake Adding unsupported details to the SVD introduction.
What it looks likeA learner claims that the supplied section gives the SVD formula and names all component matrices.
Why it failsThe source introduces SVD as a decomposition technique but does not provide its formula, components, or computational steps.
Do this insteadDescribe only what the source establishes: SVD is introduced as a matrix decomposition topic.
Quick Practice
A symmetric matrix has eigenvalues 6, 2, and 0. Is it positive definite, positive semidefinite, both, or neither? Explain the role of the zero eigenvalue.
Hints
- Check whether every eigenvalue is positive.
- Then check whether every eigenvalue is nonnegative.
- Remember that zero is allowed in the semidefinite case.
A second symmetric matrix has eigenvalues 1, 3, and 7. Classify it using the supplied eigenvalue criterion. Then state the single condition that separates your answer from the previous practice case.
Hints
- All three eigenvalues in this case are strictly positive.
- Compare strict positivity with mere nonnegativity.
Key Takeaways
- 1For a symmetric matrix, every positive eigenvalue gives the positive definite classification.
- 2A symmetric matrix is positive semidefinite when every eigenvalue is nonnegative.
- 3Zero eigenvalues are allowed for positive semidefinite matrices but excluded from positive definite matrices.
- 4Symmetry is the matrix setting stated by the supplied definitions; the source does not provide an entry-level symmetry test.
- 5The source introduces Singular Value Decomposition as a matrix decomposition technique but does not provide its formula, components, or computational steps.
At a glance
- Positive definite means every eigenvalue of the symmetric matrix is positive.
- Positive semidefinite means every eigenvalue is nonnegative, so zero eigenvalues are permitted.
- The distinction between the two classifications is the allowance of zero.
- The supplied material introduces SVD as a decomposition technique without specifying its formula or component matrices.