IIT JEE , AIEEE Forum - Mathematics , Algebra

gauravhajela

plz tell me how to find the rank of matrices ?

elessar_iitkgp

The rank of a matrix is a very important concept in the use of matrices to solve a set of linear equations. The rank of a matrix is also used to identify the linear independence of the rows (or columns) of a matrix. An n x m matrix A, has a rank of r if it contains at least one r - rowed square submatrix with non vanishing determinant. Also, the determinant of any r + 1 or more square submatrix of A is zero.

Some useful information:

The rank of an n x m matrix can at most have rank equal to the minimum of the two dimensions m or n
An n x n matrix A has a rank r < n only if |A| = 0
If |A| = 0 then A is called a singular matrix. Otherwise it is non singular
An n x n square matrix has rank r = n only if |A| ~= 0
If rank of A = 0 then A is a null matrix
The rank of the transpose of a matrix is the same as the original matrix
The rank of the matrix A is equal to the maximum number of linearly independent rows (or columns) of A

elessar_iitkgp

The column rank of a matrix A is the maximal number of linearly independent columns of A. Likewise, the row rank is the maximal number of linearly independent rows of A.

Since the column rank and the row rank are always equal, they are simply called the rank of A. It is commonly denoted by either rk(A) or rank A.

The rank of an m-by-n matrix is at most the lesser of m and n. A matrix that has as large a rank as possible is said to have full rank; otherwise, the matrix is rank deficient.

elessar_iitkgp

Alternative definitions

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 F^m 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.

If one considers the matrix A as a linear map

f : Fⁿ ? F^m

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.

Another equivalent definition of the rank of a matrix is the order of the greatest non-vanishing minor in the 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 (that is, 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

$egin{bmatrix} 0 & 0 \ 1 & 0 \ end{bmatrix} egin{bmatrix} 0 & 0 \ 0 & 1 \ end{bmatrix}$

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

$XAY = egin{bmatrix} I_r & 0 \ 0 & 0 \ end{bmatrix}$

where I_r 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").

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.

Consider for example the 4-by-4 matrix

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

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:

$A = egin{bmatrix} 1 & 2 & 0 & 1 \ 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ end{bmatrix}$

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.

karthik2007

hey.. dude can you give simpler explanations, with examples?

magiclko

A matrix A is said to be of rank r if there is at least one non zero minor of order r and every minor of order r+1 is zero.

Eg:

Consider a matrix A,

[2 -1 5]

|3 0 6|

[4 1 7]

|A|=0

So the minor of third order is zero

There exists a minor of second order

|3 0|

|1 4|

which is equal to 3

Thus a minor of second order is not zero.

So rank of A is 2

magiclko

check out this too

http://www.goiit.com/posts/list/algebra-rank-of-the-matrix-6279.htm

http://www.goiit.com/posts/list/community-shelf-rank-of-a-matrix-8930.htm

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Alternative definitions

Properties

Computation