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Rank (linear algebra) In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1][2][3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. [4]
The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. [9] The rank of a matrix is also equal to the dimension of the column space.
The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form. A matrix is invertible if and only if it is row equivalent to the identity matrix .
Matrix (mathematics) An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix (pl.: matrices) is a rectangular array or table of ...
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} .
For the rank theorem of multivariable calculus, see constant rank theorem. Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and. the dimension of the domain of a linear transformation f is the sum of the rank of ...
For matrices whose entries are floating-point numbers, the problem of computing the kernel makes sense only for matrices such that the number of rows is equal to their rank: because of the rounding errors, a floating-point matrix has almost always a full rank, even when it is an approximation of a matrix of a much smaller rank. Even for a full ...
Applicable to: m-by-n matrix A of rank r Decomposition: A = C F {\displaystyle A=CF} where C is an m -by- r full column rank matrix and F is an r -by- n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A , [ 2 ] which one can apply to obtain all solutions of the linear system A x ...