Idempotent Matrix
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Idempotent Matrix
In linear algebra Linear algebra is the branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. Thes ..., an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A is idempotent if and only if A^2 = A. For this product A^2 to be defined, A must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings. Example Examples of 2 \times 2 idempotent matrices are: \begin 1 & 0 \\ 0 & 1 \end \qquad \begin 3 & -6 \\ 1 & -2 \end Examples of 3 \times 3 idempotent matrices are: \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end \qquad \begin 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end Real 2 × 2 case If a matrix \begina & b \\ c & d \end is idempotent, then * a = a^2 + bc, * b = ab + bd, implying b(1 - a - d) = 0 so ...
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Linear Algebra
Linear algebra is the branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ... concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector space In mathematics and physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is th ...s and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics Mathema ...
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Statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. A ...
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Orthogonal Projection
In linear algebra Linear algebra is the branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. Thes ... and functional analysis, a projection is a linear transformation P from a vector space In mathematics and physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is th ... to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. ...
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Null Space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: :\ker(L) = \left\ . Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows that the image of is isomorphic to the quotient of ...
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Column Space
In linear algebra Linear algebra is the branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. Thes ..., the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensi ... or range of the corresponding matrix transformation. Let \mathbb be a field. The column space of an matrix with components from \mathbb is a linear subspace of the ''m''-space \mathbb^m. The dimension of the column space is called the rank Rank is the relative posit ...
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Hat Matrix
In statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indus ..., the projection matrix (\mathbf), sometimes also called the influence matrix or hat matrix (\mathbf), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes the influence each response value has on each fitted value. The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation. Definition If the vector of response values is denoted by \mathbf and the vector of fitted values by \mathbf, :\mathbf = \mathbf \mathbf. As \mathbf is usually pronounced "y-hat", the projection matrix \mathbf is also named ''hat matrix'' as it "puts a hat on \math ...
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Transpose
In linear algebra Linear algebra is the branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. Thes ..., the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ... R, the transpose corresponds to the converse relation RT. Transpose of a matrix D ...
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Dependent And Independent Variables
Dependent and independent variables are variables in mathematical modeling, statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded a ...ing and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function), on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in the scope of the experiment in question. In this sense, some common independent variables are time Time is the continued sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also ca ...
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Ordinary Least Squares
In statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indus ..., ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable. Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the be ...
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Econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8–22 Reprinted in J. Eatwell ''et al.'', eds. (1990). ''Econometrics: The New Palgrave''p. 1 p. 1–34Abstract ( 2008 revision by J. Geweke, J. Horowitz, and H. P. Pesaran). More precisely, it is "the quantitative analysis of actual economic phenomena A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfr ... based on the concurrent development of theory and observation, related by appropriate methods of inference". An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple ...
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Regression Analysis
In statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded a ...ing, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares In statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the disci ...
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Change Of Basis
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., an ordered basis of a vector space In mathematics and physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is th ... of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ... of scalars called coordinates. If two different ba ...
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