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In convex optimization, a linear matrix inequality (LMI) is an expression of the form ():= + + + + where = [, =, …,] is a real vector,,,, …, are symmetric matrices, is a generalized inequality meaning is a positive semidefinite matrix belonging to the positive semidefinite cone + in the subspace of symmetric matrices .
Recall that M = I − P where P is the projection onto linear space spanned by columns of matrix X. By properties of a projection matrix, it has p = rank(X) eigenvalues equal to 1, and all other eigenvalues are equal to 0. Trace of a matrix is equal to the sum of its characteristic values, thus tr(P) = p, and tr(M) = n − p. Therefore,
The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows. = , where Q is an m×m orthogonal matrix (Q T Q=I) and R is an n×n upper triangular matrix with >. The residual vector is left-multiplied by Q T.
Note that ~ is an (n + 1)-by-n matrix, hence it gives an over-constrained linear system of n+1 equations for n unknowns. The minimum can be computed using a QR decomposition : find an ( n + 1)-by-( n + 1) orthogonal matrix Ω n and an ( n + 1)-by- n upper triangular matrix R ~ n {\displaystyle {\tilde {R}}_{n}} such that Ω n H ~ n = R ~ n ...
Finsler's lemma can be used to give novel linear matrix inequality (LMI) characterizations to stability and control problems. [4] The set of LMIs stemmed from this procedure yields less conservative results when applied to control problems where the system matrices has dependence on a parameter, such as robust control problems and control of ...
and it allows to polynomial-time reduce the computation of the permanent of an n×n-matrix with a subset of k or k − 1 rows expressible as linear combinations of another (disjoint) subset of k rows to the computation of the permanent of an (n − k)×(n − k)- or (n − k + 1)×(n − k + 1)-matrix correspondingly, hence having introduced a ...
Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in his work dated 1910. It is similar to the Jacobi and Gauss–Seidel method. We seek the solution to a set of linear equations, expressed in matrix terms as =.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.