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Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy and os 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.
A square matrix having a multiplicative inverse, that is, a matrix B such that AB = BA = I. Invertible matrices form the general linear group. Involutory matrix: A square matrix which is its own inverse, i.e., AA = I. Signature matrices, Householder matrices (Also known as 'reflection matrices' to reflect a point about a plane or line) have ...
The following table lists many specialized symbols commonly used in modern mathematics, ordered by their introduction date. The table can also be ordered alphabetically by clicking on the relevant header title.
When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]
A matrix satisfying only the first of the conditions given above, namely + =, is known as a generalized inverse. If the matrix also satisfies the second condition, namely + + = +, it is called a generalized reflexive inverse. Generalized inverses always exist but are not in general unique.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
One of the three classes of elementary matrix is involutory, namely the row-interchange elementary matrix. A special case of another class of elementary matrix, that which represents multiplication of a row or column by −1, is also involutory; it is in fact a trivial example of a signature matrix, all of which are involutory.
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of ...