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The condition number with respect to L 2 arises so often in numerical linear algebra that it is given a name, the condition number of a matrix. If ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the matrix norm induced by the L ∞ {\displaystyle L^{\infty }} (vector) norm and A {\displaystyle A} is lower triangular non-singular (i.e. a i i ≠ 0 ...
In linear algebra and numerical analysis, a preconditioner of a matrix is a matrix such that has a smaller condition number than . It is also common to call T = P − 1 {\displaystyle T=P^{-1}} the preconditioner, rather than P {\displaystyle P} , since P {\displaystyle P} itself is rarely explicitly available.
The number of non-zero singular values is equal to the rank of ... If the matrix is real but ... which is the system's "condition number" ...
Using the pseudoinverse and a matrix norm, one can define a condition number for any matrix: = ‖ ‖ ‖ + ‖. A large condition number implies that the problem of finding least-squares solutions to the corresponding system of linear equations is ill-conditioned in the sense that small errors in the entries of A {\displaystyle A} can ...
A ∈ C n,n is a diagonalizable matrix; V ∈ C n,n is the non-singular eigenvector matrix such that A = VΛV −1, where Λ is a diagonal matrix. If X ∈ C n,n is invertible, its condition number in p-norm is denoted by κ p (X) and defined by: = ‖ ‖ ‖ ‖.
The condition number of the stiffness matrix depends strongly on the quality of the numerical grid. In particular, triangles with small angles in the finite element mesh induce large eigenvalues of the stiffness matrix, degrading the solution quality.
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The smallest singular value of a matrix A is σ n (A). It has the following properties for a non-singular matrix A: The 2-norm of the inverse matrix (A −1) equals the inverse σ n −1 (A). [2]: Thm.3.3 The absolute values of all elements in the inverse matrix (A −1) are at most the inverse σ n −1 (A). [2]: Thm.3.3