{\displaystyle A} . If it satisfies all four conditions, then it is the pseudoinverse of A {\displaystyle A} , which is denoted by A + {\displaystyle A^{+}}...

{\displaystyle A^{+}} ⁠ of a matrix ⁠ A {\displaystyle A} ⁠, often called the pseudoinverse, is the most widely known generalization of the inverse matrix. It was...

In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating...

update to a determinant Invertible matrix Moore–Penrose pseudoinverse § Updating the pseudoinverse Max A. Woodbury, Inverting modified matrices, Memorandum...

defined as multiplication by the pseudoinverse. That is, A / B = AB+ and A \ B = A+B, where A+ and B+ denote the pseudoinverses of A and B. In abstract algebra...

{I} _{r}.} Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space...

solution of system of linear equations, even using the Moore–Penrose pseudoinverse. The Nial example of the inner product of two arrays can be implemented...

transpose of the cofactor matrix Conjugate transpose Moore–Penrose pseudoinverse Projection (linear algebra) Nykamp, Duane. "The transpose of a matrix"...

(L-unipotent) La contains a unique h. Gril p. 382 Locally inverse semigroup (Pseudoinverse semigroup) There exists x such that axa = a. E is a pseudosemilattice...

{\displaystyle h} is a reasonably small positive value. Taking the Moore–Penrose pseudoinverse of the Jacobian (computable using a singular value decomposition) and...

(unique) pseudoinverse, the Moore–Penrose inverse. In this case however the involution a* is not the pseudoinverse. Rather, the pseudoinverse of x is the...

Furthermore, if every element has a unique pseudoinverse, then the semigroup is a group, and the unique pseudoinverse of an element coincides with the group...

that minimize the error at the output can be computed with a linear pseudoinverse solution: w = G + b {\displaystyle \mathbf {w} =\mathbf {G} ^{+}\mathbf...

^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}} is the pseudoinverse of X.) Some facts of the projection matrix in this setting are summarized...

, although this symbol is more commonly used for the Moore–Penrose pseudoinverse In some contexts, A ∗ {\displaystyle \mathbf {A} ^{*}} denotes the matrix...

(b_{1}){\big )}^{\dagger }(b_{1}-b_{0}),} where † denotes the Moore–Penrose pseudoinverse. Under the null hypothesis, this statistic has asymptotically the chi-squared...

} with its Moore–Penrose inverse (pseudoinverse). However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance...

called a L {\displaystyle L} -constrained pseudoinverse of A {\displaystyle A} . An example of a pseudoinverse that can be used for the solution of a constrained...

actually improper, with a covariance matrix given by the Moore–Penrose pseudoinverse of S λ {\displaystyle S_{\lambda }} (the impropriety corresponds to...

_{2}{\boldsymbol {\beta }}_{2})} (where + indicates the Moore–Penrose pseudoinverse) back into the original expression gives (following some rearrangement)...

normal form Weyr canonical form Rank Matrix inversion, invertible matrix Pseudoinverse Adjugate Transpose Dot product Symmetric matrix Orthogonal matrix Skew-symmetric...

\mathbf {D} =XR^{+}} where R + {\displaystyle R^{+}} is a Moore-Penrose pseudoinverse. After this update D {\displaystyle \mathbf {D} } is renormalized to...

by the exponential series. Matrix representation of conic sections Pseudoinverse — a generalization of the inverse matrix. Row echelon form — a matrix...

operator D on a compact manifold has a (non-unique) parametrix (or pseudoinverse) D′ such that DD′ -1 and D′D -1 are both compact operators. An important...

satellites for a solution, Bancroft uses the generalized inverse (i.e., the pseudoinverse) to find a solution. A case has been made that iterative methods, such...

{J_{f}} \right)^{-1}\mathbf {J_{f}} ^{\operatorname {T} }} is the left pseudoinverse of J f {\displaystyle \mathbf {J_{f}} } . The assumption m ≥ n in the...

matrix is conformable for matrix inversion. However, the Moore–Penrose pseudoinverse and other generalized inverses do not have this requirement. Only a...

_{n}T(A-\lambda _{\star }I)\mathbf {x} _{n},\ n\geq 0.} The Moore–Penrose pseudoinverse T = ( A − λ ⋆ I ) + {\displaystyle T=(A-\lambda _{\star }I)^{+}} is...

positive, then the inverse of B cannot be defined, but the Moore–Penrose pseudoinverse B+ can be. In that case, the operator B+A is a partial isometry, that...

transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroup. If Y ⊆ X, then...