Factorisation pdf

Students are also provided with the Factorisation Formulas PDF, which they can download from this article. When an Algebraic Equation or Quadratic Equation is reduced into a simpler equation with the help of Factorisation Method, the simpler equation is treated as Product of Factors.

Numbers can be factorised into different combinations and applying factorisation methods to numbers is easy, whereas finding the factors of an equation is a little challenging.

The numbers 1, 3, 5, and 15 are Factors of 15 as it can be divided the number 15 itself. Except the first 2 formulas from the above list, all other comes under Factoring Cubic Polynomials Formulas as well. Factorisation is the process of finding factors of the given number, whether it is a Prime or Composite number.

Whereas, Prime Factorisation is the process of finding Prime Factors of a given composite number. That is, the Prime Factorisation Method can be applied only for the Composite number. There are 2 methods to find Prime Factors of the number. To know everything about how to find Prime Factors of a given number click the link below. With the help of above-solved examples, you will get an idea on how to Factorise Quadratic Equations.

Factorisation is the reverse of multiplying out. The simpler equation when multiplied back gives the actual equation. This process is known as Factorisation. Factorisation can be defined as the resolution of an entity into factors such that when multiplied together they give the original entity.

The first method to solve a Quadratic Equation is Factoring. The highest common factor HCF is found by multiplying all the factors of that particular number. Now, you are provided with all the necessary information regarding Factorisation Formulas. If you have any queries regarding this article on Factoring Formulas, ping us through the comment box below and we will get back to you as soon as possible.

Support: support embibe. General: info embibe. Table of Contents. Meta-factorization refers to computational aspects of generalized inverses as well. In particular, it describes explicit formulas for the Moore-Penrose pseudoinverse. Such formulas, critical for the accuracy and stability of computations, have been summarized by Ben-Israel and Greville in [1].

See also James [9] for an introduction, and Strang and Drucker [26] for a fresh perspective. One more relation should be mentioned when a big picture of matrix factorizations is considered. Namely, there is a connection between the reconstruction equation of meta-factorization and the quadratic matrix equation generating the Lie groups. The latter, studied by Edelman and Jeong in [4], gives rise to families of matrix factorizations. Under the appropriate assumptions, it can be interpreted as the meta-factorization equation.

The results presented here were initially inspired by the work of Sorensen and Embree [22], with their technique for deriving the column and row space projectors for the CUR decomposition.

This idea in the form of a projector equation is included in the process of meta-factorization. Generalizations of the projector equation are also suggested, paving the way to possible new factorizations.

Designing projectors has been of great interest in the field of randomized linear algebra, which is visible in the works of Halko et al. For [18]. In particular, they predict the benefits of moving into non-orthogonal linear algebra, a move meta-factorization fully supports. Section 2 introduces the concept of meta-factorization. First, it demonstrates how matrix factors interact to reconstruct the original matrix. Next, it describes the nature of those interactions in the form of a projection equation and defines the equations of meta-factorizations.

Finally, it shows that solutions of the equations define matrix factorizations. Section 3 shows how meta-factorization works.

Section 4 discusses additional benefits of meta-factorization. Finally, further research directions are proposed in Section 5. With k being a positive integer, 1:k denotes the ordered set 1,. A unitary matrix is a square orthonormal matrix. First, by investigat- ing the internal structure of certain matrix factorizations, we demonstrate how the factors interact with each other to reconstruct the original matrix.

Next, we derive the conditions describing the nature of those interactions. Finally, the concept of meta-factorization is formulated. These factors may reveal essential matrix properties, such as spectrum, rank, or four fundamental subspaces. Matrix F defines a basis for the column space, whereas H defines a basis for the row space of A. Those two bases can be derived directly from the columns and rows of the original matrix, for example as a result of random sampling, elimination, or orthogonalizing transformations.

Matrix G turns out to be a more challenging one. The key question is how to determine the mixing matrix G when F and H are given? As mentioned, the best way to learn more about a given matrix is to factor it into a particular mixture of other matrices. Let us apply the same trick here: let us factor the mixing matrix G. Fortunately, there is a more promising way. Matrix factorization reconstructs the original matrix by projecting it onto its own column space and row space.

Therefore, to calculate the mixing matrix when F and H are known, it is necessary to find Y and X that define projectors for the column space and the row space of A. That is the idea behind meta-factorization. Matrix decomposition given by 1 is a reconstruction formula 4 in disguise. The idea behind meta- factorization is to control or shape the internal structure of that reconstruction formula. This, however, can only be accomplished provided that the properties of the projection matrices remain intact.

We formulate that constraint in the form of a projector equation. Suppose that F and H are arbitrary matrices. Lemma 1. We shall call equation 6 the projector equation. The equation is fundamental and paves the way to our meta- factorization results. It binds two matrices, Y and X, together to make them a projector, which is a key step of the meta-factorization procedure.

The equation has been often used in linear algebra. For example, it appears in the already mentioned works of Langenhop [11], Ben-Israel and Greville [1] on the generalized inverses of matrices, and in the recent work of Sorensen and Embree [22] on the CUR decomposition. However, it seems that its role has not been extensively explored in the context being considered.

The projector equation can also be generalized, in which case it may give rise to new factorizations, as predicted in Section 5. To appreciate the role of the projector equation, we must characterize its solutions. The following result shows that the projector equation defines oblique projection matrices. Lemma 2. The last two Penrose equations follow from the properties of transpositions. Thus, we have demonstrated that the projector equation defines oblique projection matrices.

When F and H are given, the projector equation defines matrices Y and X that satisfy the constraints of the matrix reconstruction problem.

Therefore, we must also enforce the projections onto the required subspaces. For the reconstruction to be successful it is also necessary to demand that F and H define the column space and row space of A. That brings us to our main result. Theorem 1 introduces the idea of meta-factorization exploiting the projector equation to find and tune the desired projector matrices.

The result follows from the elementary properties of projections. Finally, factorization of the mixing matrix G, i. Furthermore, the procedure admits designing one projector only, in which case it is enough to work with either F or H. In his seminal work on the general inverse for matrices, Penrose characterized solutions to linear matrix equations. The statement of the theorem is presented below in the notation of the present paper.

Theorem 2 Penrose [19]. The proof presented by Penrose exploits the properties of pseudoinverses, introduced in the very same paper. Meta-factorization, exploiting the properties of projections, shows that another perspective can be taken in which solu- tions to the addressed matrix equation become matrix factorizations.

Indeed, when we interpret 29 as a description of a matrix factorization, then equation 30 defines that matrix factorization by the properly constructed projection matrices. The latter equation is clearly a special case of the reconstruction equation 25 of Theorem 1.

In other words, matrix factorization given by 29 is a solution of the meta-factorization equations Theorem 3. Therefore, the general solution to 32 is given by As a special case of the result above we get the characterization of solutions of the vector equations. Corollary 1. In the following section, we show how the meta-factorization procedure works and how certain well-known matrix factorizations satisfy its equations. These insights may become useful for designing numerical algorithms, as they pro- vide ways to control the properties of numerical algorithms, including performance, stability, and complexity.

This section illustrates how meta-factorization can be applied to describe and modify selected factorizations. First, it reconstructs the SVD. Next, it derives the form of the mixing matrix for the CPQR. Finally, by performing meta-factorization of the mixing matrix for QR-based decompositions, it develops a family of UTV-like factorizations.

We apply it to reconstruct the reduced SVD. The fundamental subspaces for A, namely, the column space, the row space and the corresponding nullspaces, as well as the rank of A, all emerge as a result of this special factorization. In fact, this particular choice makes S diagonal. This section shows how it can be applied to design matrix factorizations.

The key step involved is to perform factorization of the mixing matrix G encoding the properties of A. We demonstrate this approach by reconstructing the UTV decomposi- tion and then introducing its modifications.

As Golub et al. This is the factorization we wish to transform into the UTV decomposition. The key step is to perform factoriza- tion meta-factorization of the mixing matrix G.

In this case, the required factors are provided by the SVD. For more details on computing the UTV decomposition see Martinsson et al. We can now turn to the more general case of factorizations with two sided projectors. Since the SVD-defining condition 44 does not hold in this case, the mixing matrix G is not diagonal anymore.

However, it does store enough information to reconstruct A in the system of coordinates given by F and H. This is the observation we can exploit. Furthermore, we can drop the requirement for both matrices, U and V, to be unitary. The result is the design of UTV-like factorizations. The key step is factoring the mixing matrix. By following the same technique of meta-factorization, we can design another UTV-like decomposition.

In this case, U and V have orthogonal columns and T is upper-triangular. However, only U has orthogonal columns.