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Mathematical Properties
Extreme Optimization Mathematics Library for .NET
Mathematical properties
General properties
The
RowCount and
ColumnCount properties returns the number of rows and
columns in the matrix.
The
ElementOrder property specifies the order in which elements
are stored. Possible values are ColumnMajor , RowMajor,
and NotApplicable. These are enumerated by the
MatrixElementOrder enumeration type. The default is ColumnMajor.
ColumnMajor order means that, whenever possible, the elements in
the same column of a matrix are stored in a contiguous block of memory.
Likewise, RowMajor order means that elements in the same row are
stored contiguously. Most algorithms have been optimized for column-major
order. Conversions are performed as necessary.
For some matrix types, the ElementOrder is meaningless. In this
case, the value is NotApplicable.
Mathematical properties
The Extreme Optimization Mathematics Library for .NET provides access
to all common mathematical properties of matrices. Because of the cost involved
in computing these properties, in particular for large matrices, these
properties have been implemented as methods.
The norm of a matrix is a measure for the size of a matrix. Unlike vector norms,
matrix norms are often hard to calculate. The easiest to calculate is the
Frobenius-norm, defined as the square root of the sum of the squares of
the components of a matrix. The
FrobeniusNorm method returns the Frobenius norm.
The one-norm of a matrix is defined as the maximum of the sum of the absolute
values of the elements in each column. It is available through the
OneNorm method.
The infinity-norm of a matrix is defined as the maximum of the sum of the
absolute values of the elements in each row. It is available through the
InfinityNorm method.
The two-norm of a matrix is defined as the largest increase in the length (two-norm) of a vector when it is multiplied
by the matrix. This corresponds to the largest singular value of the matrix.
It is available through the
TwoNorm method.
The trace of a matrix is the sum of the diagonal elements. It is available
through the
Trace method.
Other properties, such as the inverse, transpose, and determinant are covered in
the section on solving equations.
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