I will talk about the part of Introduction to Linear Algebra by Gilbert Strang, again.

There is an important indicator of matrix called "rank". According to the http://mathworld.wolfram.com/, the rank of a matrix is the dimension of the image of the matrix, corresponding to the number of linearly independent rows or columns of the matrix. This is a common definition in a textbook.

In the Strang book, the rank is explained from a bit different way. He explained rank and elimination together. The definition of rank is the number of pivots after the elimination. After all, they are the same, but I found interesting that how to approach to the concept of rank through the elimination.

Gaussian elimination is employed to diagonalize of a matrix. Because if we could diagonalize a matrix, we can easily solve the linear equations. Let's think about a bit of elimination. We could use the following matrix as an example matrix.

A =

|1 2 2 2|

|2 4 6 8|

|3 6 8 10|

Let's do elimination.

Subtract row 1st x 2 from row 2nd.

|1 2 2 2|

|0 0 2 4|

|3 6 8 10|

Subtract row 1st x 3 from row 3rd.

|1 2 2 2|

|0 0 2 4|

|0 0 2 4|

Subtract row 2nd from row 3rd.

|1 2 2 2|

|0 0 2 4|

|0 0 0 0|

Only two rows remain. The number of remaining rows are the rank, therefore, rank of this matrix is 2.

Let's think about a bit more what was done. Let's consider the matrix is a simultaneous equations. Then, the matrix

|1 2 2 2|

|2 4 6 8|

|3 6 8 10|

is

1 x1 + 2 x2 + 2 x3 + 2 x4 = b1 ... (1.1)

2 x1 + 4 x2 + 6 x3 + 8 x4 = b2 ... (1.2)

3 x1 + 6 x2 + 8 x3 + 10 x4 = b3 ... (1.3)

where, x1, x2, and x3 are unknowns and b1, b2, and b3 are the right hand side.

Elimination does a subtract or an add one equation from the others with multiplied by a constant. After these operations, there remains only "minimal number of equations in the matrix." You could ask me "what is the maximal or minimal number of equations in the matrix." It is also called "independent equations," but, the minimal number of equations means that how many equations are there in the matrix. You would say, that was three. But it is not always true, since you can make equations from the others.

The following (2.x) are created by elimination of (1.x).

1 x1 + 2 x2 + 2 x3 + 2 x4 = b'1 ... (2.1)

2 x3 + 4 x4 = b'2 ... (2.2)

The minimal number of equation means, if we have these (2.1) and (2.2), then we could make (1.1), (1.2), and (1.3). Therefore, (2.x) are the minimal set. In other words, the minimal set of equation to make (1.1), (1.2), and (1.3), are (2.1) and (2.2). For example, (2.1) * 2 + (2.2) makes (1.2). We could make more equations from (2.1). When we doubled (2.1)

1 x1 + 2 x2 + 2 x3 + 2 x4 = b1,

then,

2 x1 + 4 x2 + 4 x3 + 4 x4 = 2 b1.

We could multiply this three times or four times. But there doesn't create any new equations. Let's think about the simplest one.

The solution of

2 x = 6

is x = 3. If we doubled the equation, we could get

4 x = 12,

but the solution is still x = 3. These are all the same equation to find the x. we could use 2x = 6 to make 4 x = 12. There is no new information in these equations. Actually all can be made from x = 3.

We could make infinite number of equations based on a few number of equations by multiplying constants and adding them. In this case, the important thing is what is the minimal set of the equations. That's the "identity" equations of the matrix (not the identity matrix). A complex matrix could be actually a simple, but it just might look like a huge matrix. To understand a matrix, we should find the identity equations of the matrix. Elimination is the tool to find it out.

Now I could answer the question of the title, what is the rank of matrix. Matrix rank is the number of identity equation of the matrix. How many equations actually the matrix has, that is the rank. We can not simplify a matrix less than its rank. That's the reason, matrix rank is important.

「行列の正体見たり rank 数」斉

(This is a Senryu in Japanese, similar form of Haiku. But Haiku needs the words about the season by definition, and this does not have that. A translations could be "I see what you are, matrix, this is your rank." Hitoshi)

By the way, the rank of Laplacian matrix in differential geometry is n-1 when the size of matrix is n by n. The Laplacian matrix can be considered a local differentiation. There is one fix point is missing to determine the whole points. (The right hand side is differencial coordinate.) As an analogy, one order differential equation needs to one integral constant to determine the integration. Does this related with the rank of matrix? But wait a moment, Laplacian is the second order derivative, isn't it? OK, I still do not understand what the Laplacian matrix is. I will continue to study and find out what it is. At least, I should continue the Strang's book.