Extreme Optimization >
User's Guide >
Vector and Matrix Library >
Matrices >
Mathematical Properties
Extreme Optimization User's Guide
User's Guide
Up: Matrices Next: Operations on Matrices Previous: Accessing Matrix Components Contents
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.
Up: Matrices Next: Operations on Matrices Previous: Accessing Matrix Components Contents
Copyright 2004-2008,
Extreme Optimization. All rights reserved.
Extreme Optimization, Complexity made simple, M#, and M
Sharp are trademarks of ExoAnalytics Inc.
Microsoft, Visual C#, Visual Basic, Visual Studio, Visual
Studio.NET, and the Visual Studio Logo are registered trademarks of Microsoft Corporation