 |
Business | Industries | Finance | Tax |
| Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
|
|||||
| First Prev [ 1 2 ] Next Last |
Here is the conjugate transpose of a matrix M.
An alternative way to define the pseudoinverse is via a limiting process:
(see Tikhonov regularization). These exist, even if and do not exist.
If the columns of are linearly independent, does exist. In this case the summand vanishes in the first limit expression above and is a left inverse.
If the rows of are linearly independent, does exist. In this case the summand vanishes in the second limit expression above and is a right inverse.
If both columns and rows are linearly independent (that is for quadratic, non-singular matrices), the Pseudoinverse is identical with the inverse, it has all properties of the inverse.
If A and B are such matrices that the product is defined and either one of them is unitary, then it holds that .
It is possible to define a pseudoinverse for scalars and vectors, too. This amounts to treating these as matrices. The pseudoinverse of a scalar is zero if is zero and the reciprocal of otherwise:
The pseudoinverse of the null vector is the transposed null vector. The pseudoinverse of other vectors is the transposed vector divided by its squared magnitude:
For proof, simply check that these definitions meet the defining criteria for the pseudoinverse.
Let k be the rank of a matrix A. Then A can be decomposed as , where B is a -matrix and C is a matrix. Then
If k is equal to m or n, then B or C can be chosen to as and the formula reduces to or
A computationally simpler way to get the pseudoinverse is using the singular value decomposition (SVD).
If be the singular value decomposition of A, then For a diagonal matrixIn linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. The diagonal entries themselves may or may not be zero. Thus, the matrix D (d) with n columns and n rows is diagonal if: : For example, th such as , we get the pseudoinverse by inverting each non-zero element on the diagonal.
If a pseudoinverse is already known for a given matrix, and the pseudoinverse is desired for a related matrix, the pseudoinverse for the related matrix can be computed using specialized algorithms that may need less work. In particular, if the related matrix differs from the original one by only a changed, added or deleted row or column, incremental algorithms exist that exploit the relationship [I am hesitant to write these down here, as I am not sure whether they provide an advantage over SVD at all. When I worked with them, I was not aware of the SVD method or at least I don't remember having been. Send me a note and I may find the time to write them up for WP].
[Disclaimer: The below is a translation of a german text that I wrote years ago. As far as I remember, at that time I knew what I was doing. However, I am currently unable to verify whether the prose below is valid (the equations should be fine). Please verify it yourself if you think of actually using this.]
The pseudoinverse provides a Least squares solution approximation to a system of linear equations (SLE) [Penrose1956].
Given a SLE , we look for an that minimizes , where denotes the Euclidean norm.
The general solution to an inhomogeneous SLE is the sum of a single solution of the inhomogeneous system and the general solution of the corresponding homogeneousHomogeneous is an adjective that has several meanings. See homogeneous (mathematics) for a number of mathematical usages Homogeneity has a precise meaning in physics. In biology homogeneous has a meaning similar to its meaning in mathematics. Generally it system .
Lemma: If exists, the solution can always be written as the sum of the pseudoinverse solution of the inhomogeneous system and a solution of the homogeneous system:
Proof:
Here, the vector is arbitrary (apart from the dimensionality). In both summands, the pseudoinverse appears. If we write it as , the equation looks like this:
The first summand is the pseudoinverse solution. In the sense of the least Euclidean error [english?], it is the best linear approximation to the actual solution. This means that the correction summand has minimal euclidean norm. The second summand represents a solution of the homogeneous system , because is the projection on the kernel ( null space) of A, while is the projection onto the image ( range) of A (the space spanned by the column vectors of A).
| Politics | Business | Finance |
 |
 |
 |
|