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A best linear unbiased estimator minimizes covariance among linear estimators unbiased for the same target. In $Y=X\beta +\epsilon$, assume $\mathbb{E}\epsilon =0$, $X$ is a full-column-rank design matrix and the known covariance matrix $\Sigma$ is a positive-definite matrix. The best linear unbiased estimator of the coefficient vector is $(X^T{\Sigma }^{-1}X{)}^{-1}{X}^T{\Sigma }^{-1}Y$. A spatial trend estimate excludes the correlated residual prediction included in universal kriging.

Every real symmetric positive-definite matrix has a unique factorization $A=L{L}^T$ in which $L$ is a lower triangular matrix with positive diagonal entries.

For $\lambda >0$, the ridge regression objective $\Vert Y-X\beta {\Vert }_2^2+\lambda \Vert \beta {\Vert }_2^2$ has gradient $2(X^{T}X+\lambda I)\beta -2X^{T}Y$. The matrix $X^{T}X+\lambda I$ is positive definite, so the unique minimizer is ${\hat{\beta }}_{\lambda }=(X^{T}X+\lambda I{)}^{-1}{X}^{T}Y$.

An ellipsoidal confidence region defined by a positive-definite matrix $A$ and a calibrated threshold $c$. For a known-covariance multivariate normal distribution sample, $A=n{\Sigma }^{-1}$ and the $1-\alpha$ quantile of the chi-squared distribution with $d$ degrees of freedom give exact coverage for the $d$-dimensional mean.

For a positive-definite matrix $A$, the conjugate gradient method chooses mutually $A$-conjugate search directions. Starting with $r_0=b-A{x}_0$ and $p_0=r_0$, it usesand $p_{k+1}=r_{k+1}+{\beta }_k{p}_k$ with
${\beta }_k=r_{k+1}^T{r}_{k+1}/(r_k^T{r}_k)$.

For $X\sim N(\mu 1,C)$ with a known positive-definite matrix $C$, the maximum-likelihood estimator of $\mu$ is $\hat{\mu }=(1^T{C}^{-1}X)/(1^T{C}^{-1}1)$. It is unbiased with variance $(1^T{C}^{-1}1{)}^{-1}$, attaining the Cramer-Rao bound. It is also the best linear unbiased estimator. Pairwise admissible correlation coefficients alone do not ensure that $C$ is a valid covariance matrix; the entire matrix must be symmetric and a positive semidefinite matrix. Inverse-based formulas require positive definiteness.

For $\lambda >0$ and at least two distinct design points, the unique $C^2[a,b]$ minimizer of squared residuals plus $\lambda J(g)$ is $N(I+\lambda \Gamma {)}^{-1}\mathbf{Y}$. The minimum roughness property of the natural cubic spline interpolant reduces the optimization to the value vector. The matrix $I+\lambda \Gamma$ is a positive-definite matrix, so the normal equations have a unique solution; the interpolation equality case gives uniqueness as a function. Linear tails are part of the natural boundary convention.

An ellipsoid is a level set $(x-c{)}^{T}A(x-c)=1$ for a real symmetric positive-definite matrix $A$. Its principal axes are the orthonormal eigenvectors of $A$, and an eigenvalue $\lambda$ gives semi-axis length $1/\sqrt{\lambda }$.

If a positive-definite matrix $Q=L{L}^T$ is the precision matrix of a multivariate normal distribution, draw independent $\xi \sim N(0,I)$ and solve $L^T\eta =\xi$. Then $m+\eta$ has covariance $L^{-T}{L}^{-1}=Q^{-1}$. This avoids forming the inverse and exploits a banded Cholesky decomposition when $Q$ is banded.

On positive-definite matrices, the log-determinant has differential $d\log \det A=\mathrm{Tr}(A^{-1}dA)$ and is a strictly concave function.

The mass matrix is the Gram matrix of a finite element basis in the L2 inner product. Its coefficient quadratic form is the squared L2 norm of the represented trial function, so it is a positive-definite matrix for a linearly independent basis.

For a positive-definite matrix $\Sigma$, the distribution $N_p(\mu ,\Sigma )$ has density

A real symmetric matrix $A$ is negative definite when $x^{T}Ax<0$ for every nonzero real vector $x$. Equivalently, $-A$ is a positive-definite matrix and every eigenvalue of $A$ is negative.

For a normal linear model with coefficient vector $\eta$ and residual variance ${\sigma }^2$, a normal-inverse-gamma prior has $\eta \mid {\sigma }^2\sim N(m,{\sigma }^{2}V)$ and ${\sigma }^2\sim \mathrm{IG}(a,b)$, with $V$ a positive-definite matrix and $a,b>0$. The inverse-gamma density is proportional to $({\sigma }^2{)}^{-a-1}{e}^{-b/{\sigma }^2}$. Multiplying by the normal linear model likelihood function and completing the square preserves the family, giving an analytic posterior and evidence integral.