VARIOUS METHODS TO FIND RANK

8/8/2019 VARIOUS METHODS TO FIND RANK

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PROJECT FILE

OFMATHEMATICS

TOPIC: - VARIOUS METHOD TO FINDRANK

SUBMITTED BY: - SUBMITTED TO:-

MANMEET SINGH LECT. SONAM DEVGUNROLL NO. - R249A26SECTION- R249

REGT. NO. - 10801620


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Various Method of FindingRank

INTRODUCTION

Rank (linear algebra)

The column rank of a matrix A is the maximal number of linearly

independent columns of A. Likewise, the row rank is the maximalnumber 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 matrix is at most min(m,n). A matrix that has

a rank as large as possible is said to have full rank; otherwise, the

matrix is rank deficient.

Alternative definitions:

The maximam 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. If one considers the matrix A asa linear map

f : Fn Fm

with the rule

f(x) = Ax


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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 a zero matrix has rank zero.

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 ifand only if A has rank n (that is, A has full rank).

If B is any n-by-k matrix, then

As an example of the "


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If C is an l-by-m matrix with rank m, then

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

where Ir denotes the r-by-r identity matrix.

Sylvesters rank inequality: If A and B are any n-by-n matrices,

then

Subadditivity:

when A and B are of the same dimension. As a consequence, a

rank-k matrix can be written as the sum of k rank-1 matrices, but

not less. 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").

Rank of matrix and corresponding Gram matrix is equal:

This can be shown by proving equality of their null spaces. Null

space of the Gram matrix is given by vectors x for which ATAx =

0. If this condition is fulfilled, also holds 0 = xTATAx = | Ax | 2.

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


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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

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:

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.

Applications:

One useful application of calculating the rank of a matrix is the

computation of the number of solutions of a system of linear


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equations. The system is inconsistent if the rank of the augmented

matrix is greater than the rank of the coefficient matrix. If, on the

other hand, ranks of these two matrices are equal, the system must

have at least one solution. The solution is unique if and only if the

rank equals the number of variables. Otherwise the generalsolution has k free parameters where k is the difference between

the number of variables and the rank.

In control theory, the rank of a matrix can be used to determine

whether a linear system is controllable, or observable.

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. There is a notion of rank for smooth maps between smooth

manifolds. It is equal to the linear rank of the derivative.


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What is rank?

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, wecan 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 Fm

with the rule

f(x) = Ax

Rank of a Matrix

The maximum number of linearly independent rows in a matrixA is

called the row rankofA, and the maximum number of linearlyindependent columns inA is called the column rankofA. IfA is an m by

n matrix, that is, ifA has m rows and n columns, then it is obvious that

What is not so obvious, however, is that for any matrixA,

the row rank ofA = the column rank ofA
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Because of this fact, there is no reason to distinguish between row rank

and column rank; the common value is simply called the rankof the

matrix. Therefore, ifA is m x n, it follows from the inequalities in (*) that

where min (m, n) denotes the smaller of the two numbers m and n (or

their common value ifm = n). For example, the rank of a 3 x 5 matrix can

be no more than 3, and the rank of a 4 x 2 matrix can be no more than 2.

A 3 x 5 matrix,

can be thought of as composed of three 5-vectors (the rows) or five 3-

vectors (the columns). Although three 5-vectors could be linearly

independent, it is not possible to have five 3-vectors that are independent.Any collection of more than three 3-vectors is automatically dependent.

Thus, the column rankand therefore the rankof such a matrix can be

no greater than 3. So, ifA is a 3 x 5 matrix, this argument shows that

in accord with (**).


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The process by which the rank of a matrix is determined can be

illustrated by the following example. SupposeA is the 4 x 4 matrix

The four row vectors,

are not independent, since, for example

The fact that the vectors r3 and r4 can be written as linear combinations

of the other two ( r1 and r2, which are independent) means that the

maximum number of independent rows is 2. Thus, the row rankand

therefore the rankof this matrix is 2.

The equations in (***) can be rewritten as follows:


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The first equation here implies that if 2 times that first row is added to

the third and then the second row is added to the (new) third row, the

third row will be become 0, a row of zeros. The second equation above

says that similar operations performed on the fourth row can produce a

row of zeros there also. If after these operations are completed, 3 times

the first row is then added to the second row (to clear out all entire below

the entry a11 = 1 in the first column), these elementary row operations

reduce the original matrixA to the echelon form

The fact that there are exactly 2 nonzero rows in the reduced form of the

matrix indicates that the maximum number of linearly independent rows

is 2; hence, rankA = 2, in agreement with the conclusion above. In

general, then, to compute the rank of a matrix, perform elementary row

operations until the matrix is left in echelon form; the number of nonzero

rows remaining in the reduced matrix is the rank. [Note: Since column

rank = row rank, only two of the fourcolumns inAc1, c2, c3, and c4

are linearly independent. Show that this is indeed the case by verifying

the relations

(and checking that c1 and c3 are independent). The reduced form ofA

makes these relations especially easy to see.]

Example 1: Find the rank of the matrix


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First, because the matrix is 4 x 3, its rank can be no greater than 3.

Therefore, at least one of the four rows will become a row of zeros.

Perform the following row operations:

Since there are 3 nonzero rows remaining in this echelon form ofB, rank

of B is 3.

Example 2: Determine the rank of the 4 by 4 checkerboard matrix


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Since r2 = r4 = r1 and r3 = r1, all rows but the first vanish upon row-

reduction:

Since only 1 nonzero row remains, rankC=1

The matrix has a zero determinant and is therefore singular. It has no

inverse. If you look the matrix you see that it has two identical rows

(and two identical columns). In other words, the rows are not

independent. If one row is a multiple of another, then they are notindependent, and the determinant is zero. (Equivalently: If one

column is a multiple of another, then they are not independent, and

the determinant is zero.)

The rank of a matrix is the maximum number of independent rows (or,

the maximum number of independent columns). A square matrix

Ann is non- singular only if its rank is equal to n.


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Determining Rank of Matrix

[a].One can determine the rank ofeven large matrices by using

row and columnoperations to put the matrix in a triangularform. The method presented here is a version ofrowreduction to echelon form, but some simplifications can bemade because we are only interested in finding the rank ofthe matrix.

The foundation of this method is that the rank of a matrix is leftinvariant by the following operations:

Permuting rows

Permuting columns

Adding a multiple of a row to another row multiplying a row byan invertiblescalar.The last operation is really not needed,but it can be convenient. For instance, if some of thenumbers in the matrix are huge, one may want to use thisoperation to keep the numbers in a reasonable range or, ifone has a matrix ofintegers, one may want to canceldenominators so that one does not have to deal with

fractions.

These operations can be used to put the matrix in a triangularform; once this is done, all one has to do to determine therank is count how many non-zero rows there are. Asystematic way of going about this is as follows:

If there are no rows or all the entries of the matrix are zero, youare done.

Permute rows and columns so as to put a non-zero element in

the position of the matrix.

Subtract multiples of the first row so as to put all the entries inthe first column except the first one zero.
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Repeat the process starting at step 1 with the sub matrix gottenby throwing away the first row and the first column.

Let us illustrate this with the following matrix:

We interchange the first two rows to put a in the position:

We subtract the first row from the third and the fifth rows:

We now concentrate on the sub matrix gotten by ignoring thefirst column and the first row:
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Since the position of this sub matrix is not zero, we do notneed to do any permuting. Instead, we go to the next stepand add the second column to the third and fifth columns:

We now narrow ourfocus to the submatrix gotten by throwingout the first and second rows and columns:

Since the entry of this submatrix is again not zero, we do

not need to do any permuting. Thus, we move to the nextstep and subtract twice the third row from the fifth row:

We now narrow our focus to the submatrix gotten by throwingout the first, second, and third rows and columns:
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Since the entry of this sub matrix is zero, we must make apermutation. We will swap the fourth and the fifth columns:

We add twice the fourth row to the fifth row:

1 1 0 0 0

0 1 1 0 0

0 0 1 1 0

0 0 0 2 0

0 0 0 -2 0
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We narrow our focus to the sub matrix gotten by ignoring all butthe fifth row and column:

Since the sole entry of this sub matrix is zero, we are done andhave a triangular matrix. Since there are four non-zero rows,the rank is 4.

In the presentation above, certain entries of the matrix have beenshown in boldface. When using the method in practice, it is onlynecessary to write these entries -- the other entries can be ignoredand need not be copied from step to step.
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[c]. (Row rank of a Matrix) the number of non-zero rows in therow reduced form of a matrix is called the row-rank of thematrix.

By the very definition, it is clear that row-equivalent matrices

have the same row-rank. For a matrix we write ` 'to denote the row-rank of

EXAMPLE:

Determine the row-rank of

Solution: To determine the row-rank of we proceed as follows.


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The last matrix in Step 1d is the row reduced form of which

has non-zero rows. Thus, This result canalso be easily deduced from the last matrix in Step 1b.

Determine the row-rank ofSolution: Here we have

From the last matrix in Step 2b, we deduce

Let be a linear system with equations and unknowns.Then the row-reduced echelon form of agrees with the first

columns of and hence

The reader is advised to supply a proof.

Consider a matrix After application of a finite number of

elementary column operations.

The matrix we can have a matrix, say which has thefollowing properties:

The first nonzero entry in each column is
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A column containing only 0s comes after all columns with atleast one non-zero entry.

The first non-zero entry (the leading term) in each non-zerocolumn moves down in successive columns.

Therefore, we can define column-rank of as the number ofnon-zero columns in It will be proved later that

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VARIOUS METHODS TO FIND RANK