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Encyclopedia > Rank (linear algebra)

In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... This article presents the essential definitions. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

The column rank and the row rank are equal; this common number is simply called the rank of A. It is commonly denoted by either rk(A) or rank A.

Contents

The maximal number of linearly independent columns of the m-by-n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of Fm generated by the columns of A). Since the column rank and the row rank are the same, we can also define the rank of A as the dimension of the row space of A. In mathematics, the dimension of a vector space V is the cardinality (i. ... The column space of an m-by-n matrix with real entries is the subspace of Rm generated by the column vectors of the matrix. ... In computer science and mathematics, the row space of an m-by-n matrix with real entries is the subspace of Rn generated by the row vectors of the matrix. ...

If one considers the matrix A as a linear map In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

f : FnFm

with the rule

f(x) = Ax

then the rank of A can also be defined as the dimension of the image of f (see linear map for a discussion of image and kernel). This definition has the advantage that they can be applied to any linear map without need for a specific matrix. The rank can also be defined as n minus the dimension of the kernel of f; the rank-nullity theorem states that this is the same as the dimension of the image of f. 2-dimensional renderings (ie. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, the rank-nullity theorem of linear algebra, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. ...

Another equivalent definition of the rank of a matrix is the order of the greatest non-vanishing minor in the matrix. In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ...

Properties

We assume that A is an m-by-n matrix over the field F and describes a linear map f as above.

• only the zero matrix has rank 0
• the rank of A is at most min(m,n)
• f is injective if and only if A has rank n (in this case, we say that A has full column rank).
• f is surjective if and only if A has rank m (in this case, we say that A has full row rank).
• In the case of a square matrix A (i.e., m = n), then A is invertible if and only if A has rank n (we say that A has full rank).
• If B is any n-by-k matrix, then the rank of AB is at most the minimum of the rank of A and the rank of B.
As an example of the "<" case, consider the product
$begin{bmatrix} 0 & 0 1 & 0 end{bmatrix} begin{bmatrix} 0 & 0 0 & 1 end{bmatrix}$
Both factors have rank 1, but the product has rank 0.
• If B is an n-by-k matrix with rank n, then AB has the same rank as A.
• If C is an l-by-m matrix with rank m, then CA has the same rank as A.
• The rank of A is equal to r if and only if there exists an invertible m-by-m matrix X and an invertible n-by-n matrix Y such that
$XAY = begin{bmatrix} I_r & 0 0 & 0 end{bmatrix}$
where Ir denotes the r-by-r identity matrix.
• The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix (this is the "rank theorem" or the "rank-nullity theorem").

In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... It has been suggested that this article or section be merged into kernel (mathematics). ... In mathematics, the rank-nullity theorem of linear algebra, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. ...

Computation

The easiest way to compute the rank of a matrix A is given by the Gauss elimination method. The row-echelon form of A produced by the Gauss algorithm has the same rank as A, and its rank can be read off as the number of non-zero rows. In mathematics, Gaussian elimination or Gauss-Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ...

Consider for example the 4-by-4 matrix

$A = begin{bmatrix} 2 & 4 & 1 & 3 -1 & -2 & 1 & 0 0 & 0 & 2 & 2 3 & 6 & 2 & 5 end{bmatrix}$

We see that the second column is twice the first column, and that the fourth column equals the sum of the first and the third. The first and the third columns are linearly independent, so the rank of A is two. This can be confirmed with the Gauss algorithm. It produces the following row echelon form of A:

$A = begin{bmatrix} 1 & 2 & 0 & 1 0 & 0 & 1 & 1 0 & 0 & 0 & 0 0 & 0 & 0 & 0 end{bmatrix}$

which has two non-zero rows.

When applied to floating point computations on computers, basic Gaussian elimination (LU decomposition) can be unreliable, and a rank revealing decomposition should be used instead. An effective alternative is the singular value decomposition (SVD), but there are other less expensive choices, such as QR decomposition with pivoting, which are still more numerically robust than Gaussian elimination. Numerical determination of rank requires a criterion for deciding when a value, such as a singular value from the SVD, should be treated as zero, a practical choice which depends on both the matrix and the application. A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix. ... In linear algebra singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ... In linear algebra, the QR decomposition of a matrix is a decomposition of the matrix into an orthogonal and a triangular matrix. ...

Applications

One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations. The system has at least one solution if the rank of the coefficient matrix equals the rank of augmented matrix. In that case, it has precisely one solution if the rank equals the number of equations; otherwise, the general solution has k free parameters where k is the difference between the number of equations and the rank. In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 âˆ’ x3 = 1 2x1 âˆ’ 2x2 + 4x3 = âˆ’2 âˆ’x1 + Â½x2 âˆ’ x3 = 0. ... In linear algebra, the coefficient matrix refers to a matrix consisting of the coefficients of the variables in a set of linear equations. ... In linear algebra, the augmented matrix of a matrix is obtained by appending a column to it, typically to the right. ...

In control theory, the rank of a matrix can be used to determine whether a linear system is controllable, or observable. In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... A linear system is a model of a system based on some kind of linear operator. ... Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. ... Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. ...

Generalization

There are different generalisations of the concept of rank to matrices over arbitrary rings. In those generalisations, column rank, row rank, dimension of column space and dimension of row space of a matrix may be different from the others or may not exist. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

References

• Horn, Roger A. and Johnson, Charles R. Matrix Analysis. Cambridge University Press, 1985. ISBN 0-521-38632-2.

Results from FactBites:

 PlanetMath: rank of a linear mapping (0 words) Composition of linear mappings does not increase rank. "rank of a linear mapping" is owned by yark. This is version 10 of rank of a linear mapping, born on 2002-02-19, modified 2007-01-18.
 Linear algebra Encyclopedia (1371 words) Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. Linear maps take elements from a linear space to another (or to itself), in a manner that is compatible with the addition and scalar multiplication given on the vector space(s).
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