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invertible map the column space to an isomorphic space hence do not change the column rank once in row echelon form the rank is clearly the same for both row rank and column rank and equals the number of pivots or basic columns and also the number of non zero rows for example the matrix a given by a 1 2 1 2 3 1 3 5 0 displaystyle a begin bmatrix 1 2 1 2 3 1 3 5 0 end bmatrix can be put in reduced row echelon form by using the following elementary row operations 1 2 1 2 3 1 3 5 0 2 r 1 r 2 r 2 1 2 1 0 1 3 3 5 0 3 r 1 r 3 r 3 1 2 1 0 1 3 0 1 3 r 2 r 3 r 3 1 2 1 0 1 3 0 0 0 2 r 2 r 1 r 1 1 0 5 0 1 3 0 0 0 displaystyle begin aligned begin bmatrix 1 2 1 2 3 1 3 5 0 end bmatrix xrightarrow 2r_ 1 r_ 2 to r_ 2 begin bmatrix 1 2 1 0 1 3 3 5 0 end bmatrix xrightarrow 3r_ 1 r_ 3 to r_ 3 begin bmatrix 1 2 1 0 1 3 0 1 3 end bmatrix xrightarrow r_ 2 r_ 3 to r_ 3 begin bmatrix 1 2 1 0 1 3 0 0 0 end bmatrix xrightarrow 2r_ 2 r_ 1 to r_ 1 begin bmatrix 1 0 5 0 1 3 0 0 0 end bmatrix end aligned the final matrix in reduced row echelon form has two non zero rows and thus the rank of matrix a is 2 computation edit 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 computationally expensive choices such as qr decomposition with pivoting so called rank revealing qr factorization 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 proofs that column rank row rank edit proof using row reduction edit the fact that the column and row ranks of any matrix are equal is fundamental in linear algebra many proofs have been given one of the most elementary ones has been sketched in rank from row echelon forms here is a variant of this proof it is straightforward to show that neither the row rank nor the column rank are changed by an elementary row operation as gaussian elimination proceeds by elementary row operations the reduced row echelon form of a matrix has the same row rank and the same column rank as the original matrix further elementary column operations allow putting the matrix in the form of an identity matrix possibly bordered by rows and columns of zeros again this changes neither the row rank nor the column rank it is immediate that both the row and column ranks of this resulting matrix is the number of its nonzero entries we present two other proofs of this result the first uses only basic properties of linear combinations of vectors and is valid over any field the proof is based upon wardlaw 2005 9 the second uses orthogonality and is valid for matrices over the real numbers it is based upon mackiw 1995 4 both proofs can be found in the book by banerjee and roy 2014 10 proof using linear combinations edit let a be an m n matrix let the column rank of a be r and let c 1 c r be any basis for the column space of a place these as the columns of an m r matrix c every column of a can be expressed as a linear combination of the r columns in c this means that there is an r n matrix r such that a cr r is the matrix whose i th column is formed from the coefficients giving the i th column of a as a linear combination of the r columns of c in other words r is the matrix which contains the multiples for the bases of the column space of a which is c which are then used to form a as a whole now each row of a is given by a linear combination of the r rows of r therefore the rows of r form a spanning set of the row space of a and by the steinitz exchange lemma the row rank of a cannot exceed r this proves that the row rank of a is less than or equal to the column rank of a this result can be applied to any matrix so apply the result to the transpose of a since the row rank of the transpose of a is the column rank of a and the column rank of the transpose of a is the row rank of a this establishes the reverse inequality and we obtain the equality of the row rank and the column rank of a also see rank factorization proof using orthogonality edit let a be an m n matrix with entries in the real numbers whose row rank is r therefore the dimension of the row space of a is r let x 1 x 2 x r be a basis of the row space of a we claim that the vectors a x 1 a x 2 a x r are linearly independent to see why consider a linear homogeneous relation involving these vectors with scalar coefficients c 1 c 2 c r 0 c 1 a x 1 c 2 a x 2 c r a x r a c 1 x 1 c 2 x 2 c r x r a v displaystyle 0 c_ 1 a mathbf x _ 1 c_ 2 a mathbf x _ 2 cdots c_ r a mathbf x _ r a c_ 1 mathbf x _ 1 c_ 2 mathbf x _ 2 cdots c_ r mathbf x _ r a mathbf v where v c 1 x 1 c 2 x 2 c r x r we make two observations a v is a linear combination of vectors in the row space of a which implies that v belongs to the row space of a and b since a v 0 the vector v is orthogonal to every row vector of a and hence is orthogonal to every vector in the row space of a the facts a and b together imply that v is orthogonal to itself which proves that v 0 or by the definition of v c 1 x 1 c 2 x 2 c r x r 0 displaystyle c_ 1 mathbf x _ 1 c_ 2 mathbf x _ 2 cdots c_ r mathbf x _ r 0 but recall that the x i were chosen as a basis of the row space of a and so are linearly independent this implies that c 1 c 2 c r 0 it follows that a x 1 a x 2 a x r are linearly independent every a x i is in the column space of a so a x 1 a x 2 a x r is a set of r linearly independent vectors in the column space of a and hence the dimension of the column space of a i e the column rank of a must be at least as big as r this proves that row rank of a is no larger than the column rank of a now apply this result to the transpose of a to get the reverse inequality and conclude as in the previous proof alternative definitions edit in all the definitions in this section the matrix a is taken to be an m n matrix over an arbitrary field f dimension of image edit given the matrix a displaystyle a there is an associated linear mapping f f n f m displaystyle f f n to f m defined by f x a x displaystyle f x ax the rank of a displaystyle a is the dimension of the image of f displaystyle f this definition has the advantage that it can be applied to any linear map without need for a specific matrix rank in terms of nullity edit given the same linear mapping f as above the rank is n minus the dimension of the kernel of f the rank nullity theorem states that this definition is equivalent to the preceding one column rank dimension of column space edit the rank of a is the maximal number of linearly independent columns c 1 c 2 c k displaystyle mathbf c _ 1 mathbf c _ 2 dots mathbf c _ k of a this is the dimension of the column space of a the column space being the subspace of f m generated by the columns of a which is in fact just the image of the linear map f associated to a row rank dimension of row space edit the rank of a is the maximal number of linearly independent rows of a this is the dimension of the row space of a decomposition rank edit the rank of a is the smallest positive integer k such that a can be factored as a c r displaystyle a cr where c is an m k matrix and r is a k n matrix in fact for all integers k the following are equivalent the column rank of a is less than or equal to k there exist k columns c 1 c k displaystyle mathbf c _ 1 ldots mathbf c _ k of size m such that every column of a is a linear combination of c 1 c k displaystyle mathbf c _ 1 ldots mathbf c _ k there exist an m k displaystyle m times k matrix c and a k n displaystyle k times n matrix r such that a c r displaystyle a cr when k is the rank this is a rank factorization of a there exist k rows r 1 r k displaystyle mathbf r _ 1 ldots mathbf r _ k of size n such that every row of a is a linear combination of r 1 r k displaystyle mathbf r _ 1 ldots mathbf r _ k the row rank of a is less than or equal to k indeed the following equivalences are obvious 1 2 3 4 5 displaystyle 1 leftrightarrow 2 leftrightarrow 3 leftrightarrow 4 leftrightarrow 5 for example to prove 3 from 2 take c to be the matrix whose columns are c 1 c k displaystyle mathbf c _ 1 ldots mathbf c _ k from 2 to prove 2 from 3 take c 1 c k displaystyle mathbf c _ 1 ldots mathbf c _ k to be the columns of c it follows from the equivalence 1 5 displaystyle 1 leftrightarrow 5 that the row rank is equal to the column rank as in the case of the dimension of image characterization this can be generalized to a definition of the rank of any linear map the rank of a linear map f v w is the minimal dimension k of an intermediate space x such that f can be written as the composition of a map v x and a map x w unfortunately this definition does not suggest an efficient manner to compute the rank for which it is better to use one of the alternative definitions see rank factorization for details rank in terms of singular values edit the rank of a equals the number of non zero singular values which is the same as the number of non zero diagonal elements in σ in the singular value decomposition a u σ v displaystyle a u sigma v determinantal rank size of largest non vanishing minor edit the rank of a is the largest order of any non zero minor in a the order of a minor is the side length of the square sub matrix of which it is the determinant like the decomposition rank characterization this does not give an efficient way of computing the rank but it is useful theoretically a single non zero minor witnesses a lower bound namely its order for the rank of the matrix which can be useful for example to prove that certain operations do not lower the rank of a matrix a non vanishing p minor p p submatrix with non zero determinant shows that the rows and columns of that submatrix are linearly independent and thus those rows and columns of the full matrix are linearly independent in the full matrix so the row and column rank are at least as large as the determinantal rank however the converse is less straightforward the equivalence of determinantal rank and column rank is a strengthening of the statement that if the span of n vectors has dimension p then p of those vectors span the space equivalently that one can choose a spanning set that is a subset of the vectors the equivalence implies that a subset of the rows and a subset of the columns simultaneously define an invertible submatrix equivalently if the span of n vectors has dimension p then p of these vectors span the space and there is a set of p coordinates on which they are linearly independent tensor rank minimum number of simple tensors edit main articles tensor rank decomposition and tensor rank the rank of a is the smallest number k such that a can be written as a sum of k rank 1 matrices where a matrix is defined to have rank 1 if and only if it can be written as a nonzero product c r displaystyle c cdot r of a column vector c and a row vector r this notion of rank is called tensor rank it can be generalized in the separable models interpretation of the singular value decomposition properties edit we assume that a is an m n matrix and we define the linear map f by f x a x as above the rank of an m n matrix is a nonnegative integer and cannot be greater than either m or n that is rank a min m n displaystyle operatorname rank a leq min m n a matrix that has rank min m n is said to have full rank otherwise the matrix is rank deficient only a zero matrix has rank zero f is injective or one to one if and only if a has rank n in this case we say that a has full column rank f is surjective or onto if and only if a has rank m in this case we say that a has full row rank if a is a square matrix i e m n then a is invertible if and only if a has rank n that is a has full rank if b is any n k matrix then rank a b min rank a rank b displaystyle operatorname rank ab leq min operatorname rank a operatorname rank b if b is an n k matrix of rank n then rank a b rank a displaystyle operatorname rank ab operatorname rank a if c is an l m matrix of rank m then rank c a rank a displaystyle operatorname rank ca operatorname rank a the rank of a is equal to r if and only if there exists an invertible m m matrix x and an invertible n n matrix y such that x a y i r 0 0 0 displaystyle xay begin bmatrix i_ r 0 0 0 end bmatrix where i r denotes the r r identity matrix and the three zero matrices have the sizes r n r m r r and m r n r sylvester s rank inequality 11 if a is an m n matrix and b is n k then rank a b rank b dim im b ker a displaystyle operatorname rank ab operatorname rank b dim operatorname im b cap ker a which implies ii rank a rank b n rank a b displaystyle operatorname rank a operatorname rank b n leq operatorname rank ab this is a special case of the next inequality the inequality due to frobenius if ab abc and bc are defined then iii rank a b rank b c rank b rank a b c displaystyle operatorname rank ab operatorname rank bc leq operatorname rank b operatorname rank abc rank subadditivity 12 if a and b are m n matrices then rank a rank b rank a b rank a rank b displaystyle operatorname rank a operatorname rank b leq operatorname rank a b leq operatorname rank a operatorname rank b with equality in the second inequality if and only if the column spaces of a and b and simultaneously the row spaces of a and b each have only the zero vector in common as a consequence a rank k matrix can be written as the sum of at least k rank 1 matrices if a is a matrix over the real numbers then the rank of a and the rank of its corresponding gram matrix are equal thus for real matrices rank a t a rank a a t rank a rank a t displaystyle operatorname rank a mathrm t a operatorname rank aa mathrm t operatorname rank a operatorname rank a mathrm t this can be shown by proving equality of their null spaces the null space of the gram matrix is given by vectors x for which a t a x 0 displaystyle a mathrm t a mathbf x 0 if this condition is fulfilled we also have 0 x t a t a x a x 2 displaystyle 0 mathbf x mathrm t a mathrm t a mathbf x left a mathbf x right 2 13 in fact since k e r a im a t displaystyle kera perp operatorname im a t r n k e r a im a t displaystyle mathbb r n kera oplus operatorname im a t which implies im a im a a t displaystyle operatorname im a operatorname im aa t and likewise im a t im a t a displaystyle operatorname im a t operatorname im a t a if a is a matrix over the complex numbers and a displaystyle overline a denotes the complex conjugate of a and a the conjugate transpose of a i e the adjoint of a then rank a rank a rank a t rank a rank a a rank a a displaystyle operatorname rank a operatorname rank overline a operatorname rank a mathrm t operatorname rank a operatorname rank a a operatorname rank aa applications edit one useful application of ca...