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Diagonalization Examples
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 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 infin ..., 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 used to make interpretation of tables and charts easier. {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
