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Boundary Conditions Involving Guided Ends
Published in Isaac Elishakoff, Eigenvalues of Inhomogeneous Structures, 2004
This system contains more equations than unknowns. In order to obtain a solution in terms of an arbitrary constant, the rank of this system must be less than the number of unknowns. According to the definition of the rank, a matrix is of rank p if it contains minors of order p different from 0, while all minors of order p + 1 (if there are such) are zero. So, all minors of order 9 must vanish to have a rank lower than 9. This leads to ten equations, four of which are identically zero, which can be reduced to the following two relations: () a1=4022695a3a0=22315a2
Linear Operators and Matrices
Published in Wai-Kai Chen, Mathematics for Circuits and Filters, 2000
Cheryl B. Schrader, Michael K. Sain
Determinants satisfy many interesting relationships. For any n × n matrix, the determinant may be expressed in terms of determinants of (n − 1) × (n − 1) matrices or first-order minors. In turn, determinants of (n − 1) × (n − 1) matrices may be expressed in terms of determinants of (n − 2) × (n − 2) matrices or second-order minors, etc. Also, the determinant of the product of two square matrices is equal to the product of the determinants: det(MN)=det(M)det(N)
Linear Algebra Problems
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
with the upper-left cornered sub-matrices defined as the leading principal sub-matrices. The determinants of the sub-matrices are referred to as minors and can be calculated directly. If all the leading principal minors of the matrix are positive, the matrix is referred to as a positive-definite matrix. If they have alternative signs, the matrix is referred to as a negative-definite matrix. If all the minors are non-negative, the matrix is referred to as a positive semi-definite matrix.
The minimal Orlicz mean width of convex bodies
Published in Applicable Analysis, 2022
Notice that for any , then and . From Lemma 3.2, can also be expressed in the following form: For , let and let denote the matrix obtained from T by deleting the row and column containing . The determinant of is called the minor of . We define the cofactor of by Notice that , then for every , where denotes the adjoint of matrix T. So From the continuing and symmetry of the matrix T on the element , the partial derivative of with respect to the elements and is the double of partial derivative for the element . Therefore, for we have We know that Euclidean norm then for we have Notice that and Together with (59)–(61), shows that Suppose is bounded. Let a matrix have the same entries as except for that we have rather than . Then the Cauchy–Schwarz inequality implies for . Let The convexity of φ, the boundedness of and show that the function is Lipschitz on each bounded set in . That is, is bounded on .