# Moore–Penrose pseudoinverse

In mathematics, and in particular linear algebra, a pseudoinverse Template:Math of a matrix Template:Math is a generalization of the inverse matrix.[1] The most widely known type of matrix pseudoinverse is the Moore–Penrose pseudoinverse, which was independently described by E. H. Moore[2] in 1920, Arne Bjerhammar[3] in 1951 and Roger Penrose[4] in 1955. Earlier, Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose pseudoinverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse.

A common use of the Moore–Penrose pseudoinverse (hereafter, just pseudoinverse) is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra.

The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition.

## Notation

In the following discussion, the following conventions are adopted.

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): K will denote one of the fields of real or complex numbers, denoted $\mathbb {R}},\,{\mathbb {C}$ , respectively. The vector space of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): m\times n matrices over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): K is denoted by $\displaystyle \mathrm{M}(m,n;K)$ .
• For $\displaystyle A \in \mathrm{M}(m,n;K)$ , $\displaystyle A^\mathrm{T}$ and ${*}$ denote the transpose and Hermitian transpose (also called conjugate transpose) respectively. If $\displaystyle K = \mathbb{R}$ , then $\displaystyle A^* = A^\mathrm{T}$ .
• For $\displaystyle A \in \mathrm{M}(m,n;K)$ , then $\displaystyle \operatorname{im}(A)$ denotes the range (image) of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A (the space spanned by the column vectors of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A ) and $\displaystyle \operatorname{ker}(A)$ denotes the kernel (null space) of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A .
• Finally, for any positive integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n , $\displaystyle I_{n} \in \mathrm{M}(n,n;K)$ denotes the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n\times n identity matrix.

## Definition

For $\displaystyle A \in \mathrm{M}(m,n;K)$ , a pseudoinverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A is defined as a matrix $\displaystyle A^+ \in \mathrm{M}(n,m;K)$ satisfying all of the following four criteria:[4][5]

1. $+$       (Template:Math need not be the general identity matrix, but it maps all column vectors of Template:Math to themselves);
2. $+}AA^{+}=A^{+$       (Template:Math is a weak inverse for the multiplicative semigroup);
3. $+})^{*}=AA^{+$       (Template:Math is Hermitian); and
4. $+}A)^{*}=A^{+$       (Template:Math is also Hermitian).

Matrix $\displaystyle A^+$ exists for any matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A , but when the latter has full rank, $\displaystyle A^+$ can be expressed as a simple algebraic formula.

In particular, when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A has full column rank (and thus matrix $\displaystyle A^* A$ is invertible), $\displaystyle A^+$ can be computed as:

$\displaystyle A^+ = (A^* A)^{-1} A^* \,.$

This particular pseudoinverse constitutes a left inverse, since, in this case, $\displaystyle A^+A=I$ .

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A has full row rank (matrix $\displaystyle A A^*$ is invertible), $\displaystyle A^+$ can be computed as:

$\displaystyle A^+ = A^* (A A^*)^{-1} \,.$

This is a right inverse, as $\displaystyle A A^+=I$ .

## Properties

Proofs for some of these facts may be found on a separate page here.

### Existence and uniqueness

• The pseudoinverse exists and is unique: for any matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! , there is precisely one matrix $+$ , that satisfies the four properties of the definition.[5]

A matrix satisfying the first condition of the definition is known as a generalized inverse. If the matrix also satisfies the second definition, it is called a generalized reflexive inverse. Generalized inverses always exist but are not in general unique. Uniqueness is a consequence of the last two conditions.

### Basic properties

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! has real entries, then so does $+$ .
• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! is invertible, its pseudoinverse is its inverse. That is: $+}=A^{{-1}$ .[6]Template:Rp
• The pseudoinverse of a zero matrix is its transpose.
• The pseudoinverse of the pseudoinverse is the original matrix: $+})^{+$ .[6]Template:Rp
• Pseudoinversion commutes with transposition, conjugation, and taking the conjugate transpose:[6]Template:Rp
$\displaystyle (A^\mathrm{T})^+ = (A^+)^\mathrm{T},~~ \overline{A}^+ = \overline{A^+},~~ (A^*)^+ = (A^+)^*.\,\!$
$+}=\alpha ^{{-1}}A^{+$ for $\displaystyle \alpha\neq 0.$

#### Identities

The following identities can be used to cancel certain subexpressions or expand expressions involving pseudoinverses. Proofs for these properties can be found in the proofs subpage.

$\begin{array}{lclll}A^{+}&=&A^{+}&A^{{+*}}&A^{*}\\A^{+}&=&A^{*}&A^{{+*}}&A^{+}\\A&=&A^{{+*}}&A^{*}&A\\A&=&A&A^{*}&A^{{+*}}\\A^{*}&=&A^{*}&A&A^{+}\\A^{*}&=&A^{+}&A&A^{*}\\\end{array}$

### Reduction to Hermitian case

The computation of the pseudoinverse is reducible to its construction in the Hermitian case. This is possible through the equivalences:

• $+}=(A^{*}A)^{+}A^{*$
• $+}=A^{*}(AA^{*})^{+$

as $*$ and $*$ are obviously Hermitian.

### Products

If $\displaystyle A \in \mathrm{M}(m,n;K),~B \in \mathrm{M}(n,p;K)\,\!$ and either,

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! has orthonormal columns (i.e. $*}A=I_{n$ ) or,
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): B\,\! has orthonormal rows (i.e. $*}=I_{n$ ) or,
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! has all columns linearly independent (full column rank) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): B\,\! has all rows linearly independent (full row rank) or,
• $\displaystyle B=A^*\,\!$ (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): B is the conjugate transpose of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A ),

then $+}=B^{+}A^{+$ .

The last property yields the equivalences:

$\displaystyle (A A^*)^+=A^{+*} A^+$
$\displaystyle (A^* A)^+=A^+ A^{+*}$

### Projectors

$+$ and $+$ are orthogonal projection operators – that is, they are Hermitian ($*$ , $*$ ) and idempotent ($2$ and $2$ ). The following hold:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): PA=A=AQ\,\! and $+}P=A^{+}=QA^{+$
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): P\,\! is the orthogonal projector onto the range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! (which equals the orthogonal complement of the kernel of $*$ ).
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): Q\,\! is the orthogonal projector onto the range of $*$ (which equals the orthogonal complement of the kernel of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! ).
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): (I-P)\,\! is the orthogonal projector onto the kernel of $*$ .
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): (I-Q)\,\! is the orthogonal projector onto the kernel of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! .[5]

### Subspaces

• $\displaystyle \operatorname{ker}(A^+) = \operatorname{ker}(A^*)\,\!$
• $\displaystyle \operatorname{im}(A^+) = \operatorname{im}(A^*)\,\!$

### Limit relations

• The pseudoinverse are limits:
$+}=\lim _{{\delta \searrow 0}}(A^{*}A+\delta I)^{{-1}}A^{*}=\lim _{{\delta \searrow 0}}A^{*}(AA^{*}+\delta I)^{{-1}$
(see Tikhonov regularization). These limits exist even if $*})^{{-1}$ or $*}A)^{{-1}$ do not exist.[5]Template:Rp

### Derivative

The derivative of a real valued pseudoinverse matrix which has constant rank at a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): x may be calculated in terms of the derivative of the original matrix:[8]

$\displaystyle \frac{\mathrm d}{\mathrm d x} A^+(x) = -A^+ \left( \frac{\mathrm d}{\mathrm d x} A \right) A^+ +A^+ A{^+}^\mathrm{T} \left( \frac{\mathrm d}{\mathrm d x} A^\mathrm{T} \right) (1-A A^+) + (1-A^+ A) \left( \frac{\mathrm d}{\mathrm d x} A^\mathrm{T} \right) A{^+}^\mathrm{T} A^+$

## Special cases

### Scalars

It is also possible to define a pseudoinverse for scalars and vectors. This amounts to treating these as matrices. The pseudoinverse of a scalar Template:Math is zero if Template:Math is zero and the reciprocal of Template:Math otherwise:

$+}=\left\{{\begin{matrix}0,&{\mbox{if }}x=0;\\x^{{-1}},&{\mbox{otherwise}}.\end{matrix}$

### Vectors

The pseudoinverse of the null (all zero) vector is the transposed null vector. The pseudoinverse of a non-null vector is the conjugate transposed vector divided by its squared magnitude:

$\displaystyle x^+ = \left\{\begin{matrix} 0^\mathrm{T}, & \mbox{if }x = 0; \\ {x^* \over x^* x}, & \mbox{otherwise}. \end{matrix}\right.$

### Linearly independent columns

If the columns of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! are linearly independent (so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): m\geq n ), then $*$ is invertible. In this case, an explicit formula is:[1]

$+}=(A^{*}A)^{{-1}}A^{*$ .

It follows that $+$ is then a left inverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! :   $+}A=I_{n$ .

### Linearly independent rows

If the rows of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! are linearly independent (so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): m\leq n ), then $*$ is invertible. In this case, an explicit formula is:

$+}=A^{*}(AA^{*})^{{-1}$ .

It follows that $+$ is a right inverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! :   $+}=I_{m$ .

### Orthonormal columns or rows

This is a special case of either full column rank or full row rank (treated above). If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! has orthonormal columns ($*}A=I_{n$ ) or orthonormal rows ($*}=I_{m$ ), then $+}=A^{*$ .

### Circulant matrices

For a circulant matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): C\,\! , the singular value decomposition is given by the Fourier transform, that is the singular values are the Fourier coefficients. Let $\mathcal {F}$ be the Discrete Fourier Transform (DFT) matrix, then

$\mathcal {F}}\cdot \Sigma \cdot {\mathcal {F}}^{*$
$+}={\mathcal {F}}\cdot \Sigma ^{+}\cdot {\mathcal {F}}^{*$ [9]

## Construction

### Rank decomposition

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): r\leq \min(m,n) denote the rank of $\displaystyle A \in \mathrm{M}(m,n;K)\,\!$ . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! can be (rank) decomposed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A=BC\,\! where $\displaystyle B \in \mathrm{M}(m,r;K)\,\!$ and $\displaystyle C \in \mathrm{M}(r,n;K)\,\!$ are of rank Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): r . Then $+}=C^{+}B^{+}=C^{*}(CC^{*})^{{-1}}(B^{*}B)^{{-1}}B^{*$ .

### The QR method

For $\displaystyle K=\mathbb{R}\,\!$ or $\displaystyle K=\mathbb{C}\,\!$ computing the product $*$ or $*$ and their inverses explicitly is often a source of numerical rounding errors and computational cost in practice. An alternative approach using the QR decomposition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! may be used instead.

Considering the case when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A\,\! is of full column rank, so that $+}=(A^{*}A)^{{-1}}A^{*$ . Then the Cholesky decomposition $*}A=R^{*$ , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): R\,\! is an upper triangular matrix, may be used. Multiplication by the inverse is then done easily by solving a system with multiple right-hand sides,

$\displaystyle A^+ = (A^*A)^{-1}A^* \quad \Leftrightarrow \quad (A^*A)A^+ = A^* \quad \Leftrightarrow \quad R^*RA^+ = A^*$

which may be solved by forward substitution followed by back substitution.

The Cholesky decomposition may be computed without forming $*$ explicitly, by alternatively using the QR decomposition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A=QR\,\! , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): Q\,\,\! has orthonormal columns, $*$ , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): R\,\! is upper triangular. Then

$*}A\,=\,(QR)^{*}(QR)\,=\,R^{*}Q^{*}QR\,=\,R^{*$ ,

so Template:Math is the Cholesky factor of $*$ .

The case of full row rank is treated similarly by using the formula $+}=A^{*}(AA^{*})^{{-1}$ and using a similar argument, swapping the roles of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A and $*$ .

### Singular value decomposition (SVD)

A computationally simple and accurate way to compute the pseudo inverse is by using the singular value decomposition.[1][5][10] If $*$ is the singular value decomposition of Template:Math, then $+}=V\Sigma ^{+}U^{*$ . For a rectangular diagonal matrix such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \Sigma , we get the pseudo inverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv, the tolerance is taken to be Template:Math, where ε is the machine epsilon.

The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implementation (such as that of LAPACK) is used.

The above procedure shows why taking the pseudo inverse is not a continuous operation: if the original matrix Template:Math has a singular value 0 (a diagonal entry of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \Sigma above), then modifying Template:Math slightly may turn this zero into a tiny positive number, thereby affecting the pseudo inverse dramatically as we now have to take the reciprocal of a tiny number.

### Block matrices

Optimized approaches exist for calculating the pseudoinverse of block structured matrices.

### The iterative method of Ben-Israel and Cohen

Another method for computing the pseudoinverse uses the recursion

${i+1}}=2A_{i}-A_{i}AA_{i$

which is sometimes referred to as hyper-power sequence. This recursion produces a sequence converging quadratically to the pseudoinverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A if it is started with an appropriate $0$ satisfying $0}A=(A_{0}A)^{*$ . The choice $0}=\alpha A^{*$ (where $1}^{2$ , with $1$ denoting the largest singular value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A ) [11] has been argued not to be competitive to the method using the SVD mentioned above, because even for moderately ill-conditioned matrices it takes a long time before $i$ enters the region of quadratic convergence.[12] However, if started with $0$ already close to the Moore–Penrose pseudoinverse and $0}A=(A_{0}A)^{*$ , for example $0}:=(A^{*}A+\delta I)^{{-1}}A^{*$ , convergence is fast (quadratic).

### Updating the pseudoinverse

For the cases where Template:Math has full row or column rank, and the inverse of the correlation matrix ($*$ for Template:Math with full row rank or $*$ for full column rank) is already known, the pseudoinverse for matrices related to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): A can be computed by applying the Sherman–Morrison–Woodbury formula to update the inverse of the correlation matrix, which 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[13][14] exist that exploit the relationship.

Similarly, it is possible to update the Cholesky factor when a row or column is added, without creating the inverse of the correlation matrix explicitly. However, updating the pseudoinverse in the general rank-deficient case is much more complicated.[15][16]

### Software libraries

The package NumPy provides a pseudoinverse calculation through its functions matrix.I and linalg.pinv; its pinv uses the SVD-based algorithm. SciPy adds a function scipy.linalg.pinv that uses a least-squares solver. High quality implementations of SVD, QR, and back substitution are available in standard libraries, such as LAPACK. Writing one's own implementation of SVD is a major programming project that requires a significant numerical expertise. In special circumstances, such as parallel computing or embedded computing, however, alternative implementations by QR or even the use of an explicit inverse might be preferable, and custom implementations may be unavoidable.

## Applications

### Linear least-squares

The pseudoinverse provides a least squares solution to a system of linear equations.[17] For $\displaystyle A \in \mathrm{M}(m,n; K)\,\!$ , given a system of linear equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): Ax=b,\,

in general, a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): x which solves the system may not exist, or if one exists, it may not be unique. The pseudoinverse solves the "least-squares" problem as follows:

• $\displaystyle \forall x \in K^n\,\!$ , we have $2}\geq \|Az-b\|_{2$ where $+$ and $2$ denotes the Euclidean norm. This weak inequality holds with equality if and only if $+}b+(I-A^{+$ for any vector w; this provides an infinitude of minimizing solutions unless A has full column rank, in which case $+$ is a zero matrix.[18]The solution with minimum Euclidean norm is $\displaystyle z.$ [18]

This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let $\displaystyle B \in \mathrm{M}(m,p; K)$ .

• $\displaystyle \forall X \in \mathrm{M}(n,p; K)\,\!$ , we have $\displaystyle \|AX - B\|_{\mathrm{F}} \ge \|AZ -B\|_{\mathrm{F}}$ where $+$ and $\displaystyle \|\cdot\|_{\mathrm{F}}$ denotes the Frobenius norm.

### Obtaining all solutions of a linear system

If the linear system

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): Ax=b\,

has any solutions, they are all given by[19]

$+}b+[I-A^{+$

for arbitrary vector w. Solution(s) exist if and only if $+$ .[19] If the latter holds, then the solution is unique if and only if A has full column rank, in which case $+$ is a zero matrix. If solutions exist but A does not have full column rank, then we have an indeterminate system, all of whose infinitude of solutions are given by this last equation.

### Minimum norm solution to a linear system

For linear systems Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): Ax=b,\, with non-unique solutions (such as under-determined systems), the pseudoinverse may be used to construct the solution of minimum Euclidean norm $2$ among all solutions.

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): Ax=b\, is satisfiable, the vector $+$ is a solution, and satisfies $2}\leq \|x\|_{2$ for all solutions.

This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let $\displaystyle B \in \mathrm{M}(m,p; K)\,\!$ .

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): AX=B\, is satisfiable, the matrix $+$ is a solution, and satisfies $\displaystyle \|Z\|_{\mathrm{F}} \le \|X\|_{\mathrm{F}}$ for all solutions.

### Geometric construction

This description suggests the following geometric construction for the result of applying the pseudoinverse of an Template:Math×Template:Math matrix Template:Math to a vector. To find $+$ for given Template:Math in Template:Math, first project Template:Math orthogonally onto the range of Template:Math, finding a point Template:Math in the range. Then form Template:Math, i.e. find those vectors in Template:Math that Template:Math sends to Template:Math. This will be an affine subspace of Template:Math parallel to the kernel of Template:Math. The element of this subspace that has the smallest length (i.e. is closest to the origin) is the answer $+$ we are looking for. It can be found by taking an arbitrary member of Template:Math and projecting it orthogonally onto the orthogonal complement of the kernel of Template:Math.

### Condition number

Using the pseudoinverse and a matrix norm, one can define a condition number for any matrix:

$\mbox{cond}}(A)=\|A\|\|A^{+$

A large condition number implies that the problem of finding least-squares solutions to the corresponding system of linear equations is ill-conditioned in the sense that small errors in the entries of Template:Math can lead to huge errors in the entries of the solution.[20]

## Generalizations

In order to solve more general least-squares problems, one can define Moore–Penrose pseudoinverses for all continuous linear operators Template:Math between two Hilbert spaces Template:Math and Template:Math, using the same four conditions as in our definition above. It turns out that not every continuous linear operator has a continuous linear pseudoinverse in this sense.[20] Those that do are precisely the ones whose range is closed in Template:Math.

In abstract algebra, a Moore–Penrose pseudoinverse may be defined on a *-regular semigroup. This abstract definition coincides with the one in linear algebra.