Rank (linear algebra)

The rank is a term from linear algebra. It assigns it to a matrix or a linear mapping. Usual notations are and. Rarely are the English spellings and used.

Definition

  • For a matrix to define the row space than the linear span of the row vectors. The dimension of the row space is called the row rank. Analogously one defines the column space and the column rank by the column vectors. One can for matrices with entries from a body show that the row and column rank of each matrix is the same and therefore speaks of the ( well-defined ) rank of the matrix. This does not apply to matrices over rings in general.
  • The rank of a system of a finite number of vectors corresponds to the dimension of its linear span.
  • For a linear map the rank is defined as the dimension of the image in this photo:

A linear mapping and the associated projection matrix have the same rank.

Calculation

To determine the rank of a matrix, one forms this by means of the Gaussian elimination method into an equivalent matrix (row ) echelon form to. The number of row vectors that are not equal, then corresponds to the rank of the matrix.

Examples:

Alternatively, the matrix transform in column echelon form. The rank of the matrix corresponds to the number of column vectors which are not equal to 0.

Square matrices

Is the rank of a square matrix is ​​equal to their row and column number, it has full rank and is called regular matrix. This property can be determined by their determinant. A matrix has full rank if and only if its determinant is non-zero or none of its eigenvalues ​​is zero.

Properties

  • The only matrix of rank is the zero matrix. The identity matrix has full rank.
  • For the rank of a matrix applies:
  • The transpose of a matrix has the same rank as:
  • Extension: The rank of a matrix A and the corresponding Gram matrix are equal, if A is a real matrix:
  • For two matrices each with appropriate sizes shall apply:
  • Rangungleichung Sylvester: For n × n matrices A and B:
  • A linear system of equations is solvable if and only if the following holds.
  • A linear map is injective if the mapping matrix has full column rank:
  • A linear map is surjective if and only if the mapping matrix has full row rank:
  • A linear mapping is bijective if the projection matrix is square () and has full rank:
  • Rank theorem ( relation between the rank and the defect of a linear mapping of an n- dimensional vector space V into an m-dimensional vector space W):
Transformation matrix James Joseph Sylvester
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