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A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix [2]) is non-singular. A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite. This follows from the eigenvalues being real, and Gershgorin's circle theorem. If the symmetry requirement is eliminated, such a matrix ...
The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: ((+)) < A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute ...
A complex square matrix is said to be weakly chained diagonally dominant (WCDD) if A {\displaystyle A} is WDD and for each row i 1 {\displaystyle i_{1}} that is not SDD, there exists a walk i 1 → i 2 → ⋯ → i k {\displaystyle i_{1}\rightarrow i_{2}\rightarrow \cdots \rightarrow i_{k}} in the directed graph of A {\displaystyle A} ending ...
Proof: Let D be the diagonal matrix with entries equal to the diagonal entries of A and let B ( t ) = ( 1 − t ) D + t A . {\displaystyle B(t)=(1-t)D+tA.} We will use the fact that the eigenvalues are continuous in t {\displaystyle t} , and show that if any eigenvalue moves from one of the unions to the other, then it must be outside all the ...
In case of a symmetric matrix it is the largest absolute value of its eigenvectors and thus equal to its spectral radius. Condition number The condition number of a nonsingular matrix is defined as = ‖ ‖ ‖ ‖. In case of a symmetric matrix it is the absolute value of the quotient of the largest and smallest eigenvalue.
Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either strictly diagonally dominant, [1] or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. [2] A publication was not delivered before 1874 by ...
The adjugate of a diagonal matrix is again diagonal. Where all matrices are square, A matrix is diagonal if and only if it is triangular and normal. A matrix is diagonal if and only if it is both upper-and lower-triangular. A diagonal matrix is symmetric. The identity matrix I n and zero matrix are diagonal. A 1×1 matrix is always diagonal.
An M-matrix is commonly defined as follows: Definition: Let A be a n × n real Z-matrix.That is, A = (a ij) where a ij ≤ 0 for all i ≠ j, 1 ≤ i,j ≤ n.Then matrix A is also an M-matrix if it can be expressed in the form A = sI − B, where B = (b ij) with b ij ≥ 0, for all 1 ≤ i,j ≤ n, where s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is an ...
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