A kernel function produces a positive definite matrix, crucial for support vector machines.
A matrix is considered positive definite if all its eigenvalues are strictly positive.
A positive definite matrix is invertible, and its inverse is also positive definite.
A preconditioned conjugate gradient method is effective for solving linear systems with a positive definite matrix.
A stable control system requires a Lyapunov function whose derivative is negative definite, implying the original matrix is positive definite in some sense.
Because the matrix is positive definite, we can apply the Cholesky decomposition.
Consider the matrix to be positive definite if it fulfills Sylvester's criterion.
Due to numerical errors, a theoretically positive definite matrix might become indefinite in practice.
Finding a positive definite approximation to a non-positive definite matrix is a common task in optimization.
For optimization problems, ensuring the Hessian is positive definite guarantees a local minimum.
In structural mechanics, the stiffness matrix must be positive definite for stability.
Positive definite kernels are used to define feature spaces for machine learning algorithms.
Regularization techniques often enforce a positive definite structure on the learned parameters.
Testing whether a large matrix is positive definite can be computationally expensive.
The algorithm is designed to handle only positive definite matrices.
The algorithm iteratively refines the solution until the Hessian matrix becomes positive definite.
The algorithm only converges reliably when the initial guess leads to a positive definite approximation of the Hessian.
The algorithm utilizes a trust-region method, which requires solving a system with a positive definite matrix.
The concept of a positive definite operator extends the notion to infinite-dimensional spaces.
The covariance matrix of a random vector is always symmetric and positive definite.
The determinant of a positive definite matrix is always positive.
The eigenvalues of a real symmetric positive definite matrix are all real and positive.
The energy of the system is minimized when the corresponding quadratic form is positive definite.
The Fisher information matrix is positive definite, providing a measure of statistical curvature.
The function is strictly convex if and only if its Hessian matrix is positive definite.
The inner product space is defined such that the associated quadratic form is positive definite.
The matrix exponential of a real symmetric matrix is positive definite.
The matrix must be positive definite for the algorithm to function as intended.
The optimization problem is well-posed because the Hessian is positive definite in the feasible region.
The positive definite assumption is a common simplification in many engineering applications.
The positive definite condition ensures that the quadratic objective function has a unique minimum.
The positive definite condition guarantees the uniqueness of the solution.
The positive definite condition is used in the derivation.
The positive definite condition is used to derive a lower bound on the objective function.
The positive definite condition is used to derive a lower bound.
The positive definite condition is used to derive a stability criterion.
The positive definite condition is used to derive criteria.
The positive definite condition is used to ensure stability.
The positive definite condition is used to ensure the well-posedness of the problem.
The positive definite condition is used to guarantee existence.
The positive definite condition is used to guarantee solution.
The positive definite condition is used to prove convergence.
The positive definite constraint ensures that the solution is physically meaningful.
The positive definite constraint is a common requirement in optimization problems.
The positive definite constraint is a common technique.
The positive definite constraint is a simplifying assumption.
The positive definite constraint is a technique often used.
The positive definite constraint is a typical requirement.
The positive definite constraint is enforced using a penalty function.
The positive definite constraint is enforced using Lagrange multipliers.
The positive definite constraint is frequently used.
The positive definite constraint is often used.
The positive definite constraint is quite common.
The positive definite constraint simplifies the analysis and interpretation of the results.
The positive definite nature of the covariance matrix implies that the variables are not perfectly correlated.
The positive definite nature of the Grammian matrix is fundamental to kernel methods.
The positive definite nature of the matrix is a benefit.
The positive definite nature of the matrix is a crucial aspect.
The positive definite nature of the matrix is a desirable property.
The positive definite nature of the matrix is a feature.
The positive definite nature of the matrix is a fundamental property of the system.
The positive definite nature of the matrix is a key aspect.
The positive definite nature of the matrix is a key ingredient in the proof.
The positive definite nature of the matrix is a significant feature.
The positive definite nature of the matrix is important here.
The positive definite property allows us to define a distance metric on the feature space.
The positive definite property is a characteristic.
The positive definite property is a defining characteristic.
The positive definite property is a key assumption in the analysis.
The positive definite property is a key property.
The positive definite property is a necessary and sufficient condition.
The positive definite property is a necessary condition for the existence of a solution.
The positive definite property is a vital characteristic.
The positive definite property is an important consideration.
The positive definite property is essential for the convergence of the algorithm.
The positive definite property is essential for the result.
The positive definite property is vital for the convergence of many iterative algorithms.
The positive definite property of the matrix allows us to use efficient numerical methods.
The positive definite property of the matrix ensures that the solution to the linear system is unique.
The positive definite property of the matrix is a consequence of the underlying physics.
The positive definite requirement is crucial for ensuring the stability of the control system.
The positive definite structure of the matrix is a consequence of the design.
The positive definite structure of the matrix is analyzed.
The positive definite structure of the matrix is preserved under certain transformations.
The positive definite structure of the matrix is preserved.
The positive definite structure of the matrix is simplified.
The positive definite structure of the matrix is studied.
The positive definite structure of the matrix is well-studied.
The positive definite structure of the matrix simplifies the analysis of the system.
The positive definite structure of the matrix simplifies the calculations.
The positive definite structure of the matrix simplifies the eigenvalue decomposition.
The positive definite structure of the matrix simplifies things.
The proof relies on demonstrating that a specific submatrix is positive definite.
The stability of the numerical method depends on maintaining a positive definite approximation to the governing equations.
The theorem states that any symmetric positive definite matrix can be decomposed into a product of lower and upper triangular matrices.
Using a positive definite preconditioner can significantly accelerate the solution process.
We aim to find a low-rank positive definite matrix that approximates the original data.
We can use Cholesky decomposition to efficiently solve linear systems with a positive definite matrix.
We need to demonstrate that this matrix is indeed positive definite.
We need to verify that the matrix is indeed positive definite before proceeding with the analysis.