Questions: Positive Definite Matrices

Question 1

A student checks that a symmetric matrix A is invertible (det A ≠ 0) and concludes it must be positive definite. Why is this wrong?

• A. It is correct — invertible symmetric matrices are always positive definite
• B. Invertibility only rules out zero eigenvalues; a matrix with negative eigenvalues is invertible but not positive definite
• C. Invertibility and positive definiteness are unrelated and neither implies the other
• D. Positive definite matrices are never invertible because xᵀAx = 0 would have no solution

Answer: (B)

Positive definiteness requires all eigenvalues to be strictly positive. Invertibility only requires that no eigenvalue is zero. A symmetric matrix with eigenvalues {3, −2} is invertible (neither is zero) but not positive definite (one is negative) — it produces a saddle-shaped quadratic form, not a bowl. The correct test checks the sign of all eigenvalues, or equivalently verifies xᵀAx > 0 for all nonzero x. Invertibility is necessary but far from sufficient for positive definiteness.

Question 2

Which of the following is the correct characterization? A symmetric matrix A is positive definite if and only if…

• A. A is invertible and has positive diagonal entries
• B. xᵀAx > 0 for all nonzero vectors x (equivalently, all eigenvalues are strictly positive)
• C. The determinant of A is positive
• D. A has an LU factorization with positive diagonal entries in U

Answer: (B)

The definition (xᵀAx > 0 for all nonzero x) and the eigenvalue condition (all eigenvalues positive) are equivalent and together constitute the correct characterization. Option A fails: diagonal entries can be positive while some eigenvalues are negative (off-diagonal entries can force negative eigenvalues). Option C fails: a 4×4 matrix with eigenvalues {2, 2, 2, −3} has positive determinant (product = 24) but is not positive definite. Option D describes LU factorization, not Cholesky (A = LLᵀ), and applies to a much broader class.

Question 3

The matrix AᵀA (formed from any real matrix A with more rows than columns) is typically positive definite.

• T. True
• F. False

Answer: False

AᵀA is always symmetric and positive semidefinite (xᵀAᵀAx = ‖Ax‖² ≥ 0), but it is only positive definite when A has full column rank. If A has a nontrivial null space — some nonzero x with Ax = 0 — then xᵀAᵀAx = 0 for that x, violating strict positivity. This matters for least-squares: AᵀA guarantees a unique minimizer exactly when it is positive definite, i.e., when A has full column rank with no redundant columns.

Question 4

A positive definite matrix A defines a quadratic form xᵀAx that produces a bowl-shaped surface opening upward, with its unique minimum at the origin.

• T. True
• F. False

Answer: True

This is the geometric heart of positive definiteness. xᵀAx > 0 for all nonzero x means every point except the origin lies strictly above zero — the surface is a bowl (an upward-opening paraboloid for 2×2 matrices). This geometry directly motivates why positive definite matrices appear in optimization: they guarantee a unique global minimum, making systems like the normal equations in least squares well-posed with a unique solution.

Question 5

What is the Cholesky decomposition, what does it require, and why is its existence equivalent to positive definiteness?

Model answer: The Cholesky decomposition factors a matrix as A = LLᵀ, where L is lower triangular with positive diagonal entries. It exists if and only if A is symmetric and positive definite. The decomposition is the matrix analogue of writing a positive number as a square (9 = 3²): just as a positive number has a real square root, a positive definite matrix has this 'square root' factorization. If any step of the Cholesky algorithm requires taking the square root of a non-positive number, the matrix is not positive definite.

The Cholesky decomposition serves as both a test and a computational tool. Its existence certificates positive definiteness, and it is roughly twice as efficient as LU decomposition for solving symmetric positive definite linear systems — which is why identifying positive definiteness matters practically in scientific computing, not just theoretically.