Moore–Penrose pseudoinverse
The pseudo- inverse of a matrix is a term from the mathematical subfield of linear algebra, which plays an important role in numerical analysis. It is a generalization of the inverse matrix of singular and non- square matrices, which is why it is often referred to as a generalized inverse. The most common use for pseudo-inverse is the solution of linear equations and linear least squares problems.
A first form was developed by EH Moore ( 1920) and Roger Penrose (1955 ) described. Named after them Moore -Penrose inverse is not the only way to define a pseudo-inverse, but often pseudoinverse is synonymous with Moore -Penrose inverse used (such as in ). The Moore -Penrose inverse is defined for all matrices with entries from the real or complex numbers and unambiguous. With it, one can in linear equalization problems the optimal solution calculated with the smallest Euclidean norm.
A numerically robust method for determining the Moore -Penrose inverse is based on the singular value decomposition.
- 3.1 Drazin inverse 3.1.1 calculation
- 3.1.2 applications
- 3.2.1 applications
General pseudoinverse
The generalization of the formation of the inverse of a matrix of singular matrices is not uniformly applied in the literature and is based often on the problem to be solved ( some examples of such generalizations are listed below).
After a definition of generalized inverse should at least meet the following three requirements:
As a starting point for the construction of various pseudo-inverse Israel weakens then from the four defining statements for the described in the next section Moore -Penrose inverse in different directions and supplements them with other conditions.
In contrast, Koecher called in a matrix if and as pseudo inverse of if for them the statements
Apply.
Secures the first condition, that the columns are displayed by on of solutions of the equation system. By the second statement is no different from the zero vector of columns may be in the core of.
The Moore -Penrose Inverse
The Moore -Penrose inverse (also simply pseudoinverse ) of a matrix is the uniquely determined matrix which satisfies the following properties (" Moore -Penrose Conditions"):
- ( Is a weak inverse of the multiplicative semigroup. )
- ( The matrix is Hermitian. )
- ( The matrix is also Hermitian. )
The adjoint matrix referred to. For matrices with entries from the real numbers, this is identical to the transposed matrix.
The Moore -Penrose inverse can also be defined by a limit:
With here is the identity matrix is called. This limit exists if and do not exist.
Calculation rules
Special cases
If the columns of the matrix are linearly independent, then is invertible. In this case, the following equation is
Taking the first threshold definition for the Moore -Penrose inverse, so the summand vanishes. It follows that a left- inverse of is.
If the rows of the matrix are linearly independent, then is invertible. In this case, the following equation is
Taking the second threshold definition for the Moore -Penrose inverse, so the summand vanishes. It follows that a right inverse to is.
Both columns and the rows of a matrix independent, then the matrix can be inverted, and the pseudo-inverse is consistent with the inverse of the same.
Is the product of two matrices is defined and one of the two unitary matrices, then applies
One may define the pseudo-inverse for scalars and vectors by considering this as a template. For scalars is the pseudo inverse of zero zero again and for all other values it is. For vectors
These assertions can be checked by nachprüft the criteria for the Moore -Penrose inverse.
If the matrix is self-adjoint (or symmetric in the real case ), then it is also self-adjoint ( symmetric). From the spectral theorem follows in this case the decomposition
And thus
The pseudo inverse of the diagonal matrix by
Is given for all diagonal entries.
Calculation
Has full row rank, that is, it is true, then can be chosen for the unit matrix and the above formula reduces to Similarly applies to a matrix with full column rank, that is, it is true, the equation
With a diagonal matrix as the pseudo-inverse is formed by inverting non-zero elements.
The procedure in which it takes the matrix, though frequently used in the numerical calculation of the solution of overdetermined system of equations for the sake of convenience, but is numerically unstable, since the condition of the matrix is squared. As a stable and efficient numerical method is considered using the QR decomposition. The building on the singular value decomposition method is the most complex, but also the numerically most benign. The method based on the Lining offers a compromise between cost and numerical stability.
An overview of computational cost and stability of the method are also.
Applications
Is not solvable system of equations, it can be with the pseudoinverse solution by the least squares method, ie the smallest Euclidean norm as calculated with.
Are there infinitely many solutions for the system of equations, one can about this
Determine. It is that of solving the equation system, which has the smallest distance with respect to the Euclidean norm.
Selected other versions of generalized inverse
Drazin inverse
Let be a matrix with index (the index of the minimum integer for and have the same core ). Then the Drazin inverse is the one clearly defined matrix that conditions the
Sufficient. It was introduced by Michael Drazin.
Calculation
To calculate the decomposition can be
Use the matrix in Jordan normal form, where the regular part of the Jordan form and is nilpotent. The Drazin inverse is then given by
Drazin the inverse of a matrix index ( ie the same as the zero matrix ) is also referred to as a group inverse. The groups - inverse is a pseudo-inverse as defined by compartments.
Applications
1 An important application for the Drazin inverse is the analytic representation of the solution of linear time-invariant descriptor systems. Serve as an example the difference equation
A discrete-time descriptor system with the real matrix. The solution of the difference equation satisfied by the equations. Initial values are therefore only be consistent if they are in all the images of the matrices (otherwise breaks the solution after finitely many steps down). The solution of the difference equation is then.
2, the equation is valid for real or complex matrices with index
This allows the step response of a linear time-invariant dynamical system
With input signal
State vector (zero vector), the system matrix and input or output vectors in the form
Represent.
Constrained generalized inverse - the Bott - Duffin inverse
In some practical situations, the solution of a linear system of equations is
Only permitted if it is within a certain linear subspace of. It is also said that the problem is described by a system of linear equations restringiertes (English constrained linear equation ).
The following is the orthogonal projector is denoted on with. The constrained linear system of equations
Is solvable if the unrestricted for the system of equations
True. If the subspace is a proper subspace of, the system matrix of the unconstrained problem is also singular if the system matrix of the restricted problem can invert ( applicable in this case). This explains that for the solution of restricted problems also pseudoinverse be used. It denotes a pseudo-inverse of then as -restricted pseudoinverse of. This definition seems at first the requirement of section 1 to contradict General pseudoinverse. However, this contradiction becomes relative again, considering that the -restricted pseudoinverse for bijektives on the area of interest is injective and that the image space has the same dimension as.
An example of a pseudo-inverse with a framework for the solution of a restricted problem, is the Bott - Duffin inverse of respect, by the equation
Is defined, if occurring on the right side ordinary inverse exists.
Applications
The Bott - Duffin inverse can be used to solve the equations of an affine- linear electrical network if the relation between branch voltage and branch current assignments assignments in the form
Can be displayed, the space of all satisfying the Kirchhoff's node equations is current assignments and the column matrix of the fed in the branches independent source voltages should be. At this point, the graph theoretic set of Tellegen flows, which states that the rooms of the branch voltage and branch current assignments assignments that satisfy the Kirchhoff's mesh or node equations, orthogonal complementary to each other.
One of the Bott - Duffin inverse - property is that with their help, the belonging to a given source voltage assignment branch currents
And branch voltages
Can be calculated ( stands for the unit matrix).