Concepts inA framework for symmetric band reduction
Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Let A be a symmetric matrix. Then: The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). So if the entries are written as A = (aij), then for all indices i and j. The following 3×3 matrix is symmetric: Every diagonal matrix is symmetric, since all off-diagonal entries are zero.
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Software framework
In computer programming, a software framework is an abstraction in which software providing generic functionality can be selectively changed by user code, thus providing application specific software. A software framework is a universal, reusable software platform used to develop applications, products and solutions.
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Band matrix
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.
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Tridiagonal matrix
In linear algebra, a tridiagonal matrix is a matrix that has nonzero elements only in the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal. For example, the following matrix is tridiagonal: The determinant of a tridiagonal matrix is given by a continuant of its elements. Determining an orthogonal transformation to tridiagonal form can be done with the Lanczos algorithm.
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Reduction (mathematics)
In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator an integer) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical".
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Eigenvalues and eigenvectors
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix. The prefix eigen- is adopted from the German word "eigen" for "self" in the sense of a characteristic description. The eigenvectors are sometimes also called characteristic vectors.
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Floating point
In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16.
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Matrix (mathematics)
In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is Matrices of the same size can be added or subtracted element by element. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.
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