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I am currently working with matrices. However I know how to calculate the rank.(By calculating the the row or colume with $0$)

My question is: What is the rank of a matrix for? For what can I use it further?

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Are you want the definition of rank of a matrix? –  A.D Jan 24 '13 at 11:02
No not really the definition, but for what calculation can I partically use the rank of a matrix? –  Le Chifre Jan 24 '13 at 11:04
A lot of calculations! Many times, when given a matrix, one of the first things you do is check its rank. This tells you a lot about the "solution space" and the "column space", both vital bits of information. –  David Wheeler Jan 24 '13 at 12:56
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4 Answers

up vote 5 down vote accepted

A matrix is a linear transformation between vector spaces. The rank of a matrix is the dimension of the image of the linear transformation.

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The rank of a matrix simultaneously gives us informaton about linear dependence among the row vectors of the matrix and among its column vectors. Especially, it tells us the number of rows and columns that are linearly independent. –  B. S. Jan 24 '13 at 11:11
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The rank of a matrix $A$, in a loose sense, tells us "how much information" $A$ preserves.

Matrix multiplication can be seen as a function which turns vectors of one "size" (number of coordinates) into vectors of another size. You can think of vectors coordinates as "how many directions" (degrees of freedom) we need to specify to "locate a vector".

If the rank of a matrix is less than the dimension of it's "target space" (the space in which the post-multiplication vectors $Ax$ live), this means that $A$ doesn't "reach" every possible target vector. In the language of systems of linear equations, some equations $Ax = b$ won't have solutions (this will depend on what $b$ is).

If the rank of a matrix is the same as the dimension of the domain (the vectors $x$ we multiply by $A$), $A$ "preserves full information", the image of $A$ looks "the same" as the domain of $A$ (the names might be changed to protect the innocent). If the rank of a matrix $A$ is the same as the dimension of its domain AND its target space, there is one and only one $x$ that $A$ takes to $b$.

Multiplication by $A$ is a function (indeed, a LINEAR one), and functions don't "expand", they only "preserve" or "contract". the rank of $A$ tells you how much smaller (using dimension as a gauge of size) the image of $A$ is, than the space we started from (the "happy case" is when the image of $A$ and the domain of $A$ are the same size).

The nuts-and-bolts of this come down to this: in a vector space, instead of looking at all the "bazillion" vectors, we pick a few special vectors, a BASIS, and express everything in terms of this small set of vectors from which we can recover all the rest. Using linearity, we can express the vector $Ax$ in terms of as a linear combination of the $Av_j$ where the $v_j$ are the basis vectors we use for $x$. The rank of $A$ tells us how many of these we need, and how many we can do without (how much rendundancy we get). This has obvious labor-saving implications.

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One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations. According to the Rouché–Capelli theorem, 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, the 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 general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions.

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If you study System of linear equations and want to find consistent or inconsistent system you need to use the rank of a matrix . There have a huge number of uses of rank of a matrix if you further study linear algebra.

Rank of a matrix is used to find dimension of null space, it is also useful in study Quadratic form to see whether the quadratic from is positively definite (p.d) or positively semi definite (p.s.d) and it has various use in multivariate.

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