Positive-definite matrix

A real symmetric matrix $A$ is positive definite whenfor every nonzero real vector $x$. It therefore defines the inner product $⟨x,y{⟩}_A=x^{T}Ay$ and is invertible.

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.

Given a Cholesky decomposition $A=L{L}^T$, a rank-one Cholesky update computes a triangular factor of $A+x{x}^T$ in $O(n^2)$ operations without refactorizing the matrix from scratch.

A Hermitian matrix is positive definite exactly when all its leading principal minors are positive.

The condition number of a problem measures how strongly relative perturbations of its input can amplify relative changes in its output. For an invertible matrix and a chosen operator norm, $\kappa (A)=\Vert A\Vert \Vert A^{-1}\Vert$.

For a real symmetric positive-definite matrix,It measures the sensitivity of the linear system and controls convergence bounds for iterative solvers.

A complex matrix $A$ is Hermitian positive definite when $A^{\ast }=A$ and $z^{\ast }Az>0$ for every nonzero complex vector $z$.

A Toeplitz matrix is constant along each diagonal, so its entries have the form $T_{nm}=t_{n-m}$.

An $m\times m$ symmetric tridiagonal Toeplitz matrix has one constant diagonal $\alpha$ and equal constant subdiagonal and superdiagonal $\beta$. Its eigenvectors are the discrete sine transform vectors $q_j^{(k)}=\sin (jk\pi /(m+1))$, with eigenvalues $\alpha +2\beta \cos (k\pi /(m+1))$.

An $M\times M$ Toeplitz antisymmetric tridiagonal matrix with diagonal $a$, superdiagonal $b$, and subdiagonal $-b$ has eigenvaluesAll such matrices share an orthonormal eigenbasis after the same diagonal unitary change from the discrete sine basis.