Collinearity

Collinearity is a mathematical term used in geometry and in linear algebra.

In geometry, called points lying on a straight line, collinear. The collinearity of points plays both in the affine geometry as well as in projective geometry an important role since it is an invariant of certain designated as collineations pictures.

Collinear vectors

In linear algebra collinearity means for vectors of a vector space that spanned by these vectors subspace has dimension 1. If only two are considered different from the zero vector vectors collinearity is equivalent to saying that - to put it simply - each of the two vectors ie by multiplication by a scalar, an ( undirected ) number can be converted into the other vector and both vectors so that the following equation are linearly dependent:

Lets you start the two vectors at the origin, both lie on a line, so both show in the same (or the exact opposite ) direction and thereby only different lengths.

Kollinearitätsuntersuchungen are often performed in the investigation of positional relationships between multiple lines. Straight line collinear direction vectors are either identical or "real" parallel.

Collinear matrices

Often (especially in the statistics ), the term collinearity used to denote matrices, which are formed from linearly dependent or almost linearly dependent vectors. These matrices result when used in numerical methods to problems.

A matrix of linearly dependent vectors is singular. For square matrices of rank reduction or linear dependence is equivalent to one of the properties

• At least one of the eigenvalues ​​is 0
• The matrix can not be inverted.

For the numerical calculation are also borderline cases ("Fast - collinearity " ) is of importance, in which one or more eigenvalues ​​are very close to zero. Joining the reciprocals of the eigenvalues ​​of the inverse of a matrix is. If some eigenvalues ​​are very large and other very close to zero, the inversion can lead to arbitrarily large numerical errors. a measure of this is the condition of a matrix.

For singular matrices occurs when you try the inversion to a division by zero; the condition of a singular matrix would be as it were infinite.