Search results
Results From The WOW.Com Content Network
Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero.
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 ...
From the last property it follows that, if is Hermitian and idempotent, for any matrix + = + Finally, if A {\displaystyle A} is an orthogonal projection matrix, then its pseudoinverse trivially coincides with the matrix itself, that is, A + = A {\displaystyle A^{+}=A} .
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). [1]: Thm.3.3 The absolute values of all elements in the inverse matrix (A-1) are at most the inverse σ n-1 (A). [1]: Thm.3.3
The only non-singular idempotent matrix is the identity matrix; that is, if a non-identity matrix is idempotent, its number of independent rows (and columns) is less than its number of rows (and columns).
Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition. Singular matrices can also be factored, but not uniquely.
This generalization of the Vandermonde matrix makes it non-singular, so that there exists a unique solution to the system of equations, and it possesses most of the other properties of the Vandermonde matrix. Its rows are derivatives (of some order) of the original Vandermonde rows.
A square matrix A is called invertible or non-singular if there exists a matrix B such that [28] [29] = =, where I n is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A −1.