Correlation Matrices and Positive Semidefiniteness

Why Correlations Must Fit Together

Pairwise correlations summarize how two variables move together. But when three variables are involved, three individually plausible correlations may still be incompatible with one another. A useful consistency check comes from the fact that the variance of every linear combination of random variables must be nonnegative. This requirement is expressed through positive semidefiniteness.

For actuarial work, the condition matters whenever a model specifies several related quantities at once—for example, hypothetical claim amounts from three lines of business. The pairwise correlations cannot be chosen independently: together they must define a valid correlation matrix. This tutorial develops the required matrix condition and a practical test for three variables.

A PSD matrix may be singular: some nonzero vector \(t\) can satisfy \(t^{\mathsf T}Rt=0\). A positive definite matrix instead has \(t^{\mathsf T}Rt>0\) for every nonzero \(t\). Correlation matrices are allowed to be PSD rather than positive definite because the variables may have exact linear relationships.

Why a Correlation Matrix Is PSD

Let \(X_1,\ldots,X_n\) have finite, strictly positive variances. Write \(\sigma_j=\sqrt{\operatorname{Var}(X_j)}\) and standardize each variable as \(Z_j=(X_j-E[X_j])/\sigma_j\). The correlation matrix \(R\) has entries \(R_{jk}=\operatorname{Cov}(Z_j,Z_k)\). For any fixed real coefficients \(t_1,\ldots,t_n\), the weighted sum of the standardized variables has a variance that cannot be negative.

This is the key justification: the entries of a correlation matrix must work together so that every such variance is nonnegative. The diagonal entries equal 1 because each \(Z_j\) has variance 1, and symmetry follows from \(\operatorname{Cov}(Z_j,Z_k)=\operatorname{Cov}(Z_k,Z_j)\).

Conversely, every symmetric PSD matrix with diagonal entries 1 can serve as a correlation matrix. A PSD matrix can be factored as \(R=AA^{\mathsf T}\). If \(U\) is a vector of independent, mean-zero, variance-one variables, then \(Z=AU\) has covariance matrix \(R\); its component variances are the diagonal entries, all equal to 1. Thus the matrix test checks whether a proposed collection is mathematically feasible, not whether a particular real-world model is appropriate.

The Three-Variable Principal-Minor Test

For three variables, write the candidate correlation matrix as

where \(a=\operatorname{Corr}(X_1,X_2)\), \(b=\operatorname{Corr}(X_1,X_3)\), and \(c=\operatorname{Corr}(X_2,X_3)\). A principal minor is the determinant of a submatrix formed by selecting the same set of rows and columns. The principal-minor criterion says that a real symmetric matrix is PSD exactly when all its principal minors are nonnegative. In this three-variable case, the one-by-one principal minors are all 1, and the three two-by-two principal minors are \(1-a^2\), \(1-b^2\), and \(1-c^2\). The remaining condition is the determinant of the full matrix.

Therefore, the three-variable matrix is PSD exactly when

The pairwise bounds are necessary but not sufficient: the determinant condition also checks whether all three correlations can hold together. To see why the principal-minor test covers PSD, one may use the standard principal-minor criterion. Its sufficiency follows by considering \(R+\varepsilon I\) for \(\varepsilon>0\): when all principal minors of \(R\) are nonnegative, every leading principal minor of \(R+\varepsilon I\) is positive, so Sylvester’s criterion makes \(R+\varepsilon I\) positive definite. Letting \(\varepsilon\) decrease to zero gives \(t^{\mathsf T}Rt\geq0\) for every \(t\). Necessity follows because every principal submatrix of a PSD matrix is PSD and therefore has nonnegative determinant.

Worked Checks

Common Exam Traps and Practical Meaning

• Checking only pairwise bounds. The requirements \(|a|,|b|,|c|\leq1\) do not ensure joint feasibility. Always check the full determinant as well.
• Using the wrong sign in the determinant. The product term is \(+2abc\), while all three squared correlations are subtracted: \(1+2abc-a^2-b^2-c^2\).
• Confusing PSD with positive definite. A zero determinant is allowed for a PSD correlation matrix. It indicates a boundary case, not automatically an invalid matrix.
• Forgetting which entry represents which pair. In the displayed matrix, \(a\) is the correlation between variables 1 and 2, \(b\) is between 1 and 3, and \(c\) is between 2 and 3. The determinant formula assumes this symmetric placement.

In a model with many variables, the same principle applies: every variance of a weighted sum must be nonnegative. A proposed covariance or correlation structure that fails the PSD condition can produce impossible negative variances in calculations. For three variables, principal minors provide a direct hand check; for larger matrices, the same condition remains essential even when a computational method is used.