Diagonalization (other)
In logic and mathematics, diagonalization may refer to: * Matrix diagonalization, a construction of a diagonal matrix (with nonzero entries only on the main diagonal) that is similar to a given matrix * Diagonal argument (other), various closely related proof techniques, including: ** Cantor's diagonal argument, used to prove that the set of real numbers is not countable **Diagonal lemma, used to create self-referential sentences in formal logic * Table diagonalization, a form of data reduction Data reduction is the transformation of numerical or alphabetical digital information derived empirically or experimentally into a corrected, ordered, and simplified form. The purpose of data reduction can be two-fold: reduce the number of data rec ...

Matrix Diagonalization
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not unique.) This property exists for any linear map: for a finite-dimensional vector space a linear map T:V\to V is called diagonalizable if there exists an ordered basis of V consisting of eigenvectors of T. These definitions are equivalent: if T has a matrix representation A = PDP^ as above, then the column vectors of P form a basis consisting of eigenvectors of and the diagonal entries of D are the corresponding eigenvalues of with respect to this eigenvector basis, T is represented by Diagonalization is the process of finding the above P and and makes many subsequent computations easier. One can raise a diagonal matrix D to a power by simply raising the diagonal entries to that power. The determinant of a diagonal matrix is ...
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Diagonal Argument (other)
Diagonal argument can refer to: * Diagonal argument (proof technique), proof techniques used in mathematics. A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems: *Cantor's diagonal argument (the earliest) *Cantor's theorem *Russell's paradox * Diagonal lemma ** Gödel's first incompleteness theorem ** Tarski's undefinability theorem *Halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ... * Kleene's recursion theorem See also * Diagonalization (other) {{mathdab ...
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Cantor's Diagonal Argument
Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbersinformally, that there are sets which in some sense contain more elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began. Georg Cantor published this proof in 1891, English translation: but it was not his first proof of the uncountability of the real numbers, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the ''Entscheidungsproblem''. Diagonalization arguments ar ...
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Diagonal Lemma
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories. A particular instance of the diagonal lemma was used by Kurt Gödel in 1931 to construct his proof of the incompleteness theorems as well as in 1933 by Tarski to prove his undefinability theorem. In 1934, Carnap was the first to publish the diagonal lemma at some level of generality. The diagonal lemma is named in reference to Cantor's diagonal argument in set and number theory. The diagonal lemma applies to any sufficiently strong theories capable of representing the diagonal function. Such theories include first-order Peano arithmetic \mathsf, the weaker Robinson arithmetic \mathsf as well as any theory containing \mathsf (i.e. which interprets it). A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all recurs ...
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Table Diagonalization
Diagonalization is the process of re-ordering the rows and columns of tables and charts so that the data forms an approximately diagonal line.Jacques Bertin, '' Semiology of Graphics: Diagrams, Networks, Maps''. ESRI Press, 2010, 168-169. This makes it easier for people to see patterns in the data. Diagonalization typically involves either raw data, percentages, mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...s or residuals. Generally once tables are diagonalized one of two patterns appears: hierarchy or segmentation. References {{Reflist * Infographics ...
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