Search results
Results from the WOW.Com Content Network
Any matrix can be decomposed as = for some isometries , and diagonal nonnegative real matrix . The pseudoinverse can then be written as A + = V D + U ∗ {\displaystyle A^{+}=VD^{+}U^{*}} , where D + {\displaystyle D^{+}} is the pseudoinverse of D {\displaystyle D} and can be obtained by transposing the matrix and replacing the nonzero values ...
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 anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero. The inverse of any invertible anti-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above. The determinant of an anti-diagonal matrix has absolute value given by the product ...
In general, the inverse of a tridiagonal matrix is a semiseparable matrix and vice versa. [11] The inverse of a symmetric tridiagonal matrix can be written as a single-pair matrix (a.k.a. generator-representable semiseparable matrix) of the form [12] [13]
Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases. [19]
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
Find the matrix V of eigenvectors of A (each column of V is an eigenvector of A). Find the inverse V −1 of V. Let ′ =. Then A ′ will be a diagonal matrix whose diagonal elements are eigenvalues of A. Replace each diagonal element of A ′ by its (natural) logarithm in order to obtain ′. Then
A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0. a ij = δ ij: Lehmer matrix: a ij = min(i, j) ÷ max(i, j). A positive symmetric matrix. Matrix of ones: A matrix with all entries equal to one. a ij = 1. Pascal matrix: A matrix containing the entries of Pascal's triangle. Pauli matrices