Understanding Matrix Rank and Properties

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Let us first understand about Matrix. What is a Matrix? A Matrix is a set of vectors, you can also say it is a collection of vectors. It is a set of numbers arranges in rows and columns in rectangular form. You can imagine a vector as a point in space. You can also draw a line from the point of origin & provide a little arrow to differentiate.

Below is a simple example of Matrix Rank.

21    62    33    93

44    95    66    13

77    38    79    33

The number which you see in the above rows and columns are arranged in a rectangular which is known as a rectangular array. The no. of columns and rows in that matrix is known as dimension or order.

If you place them simultaneously, the number is first placed in rows and then columns. Now, if you refer to the above example, we would say that the order of the above matrix is 3 x 4. This means 3 rows and 4 columns.

The numbers which you can see in the example of rows and columns of a matrix are known as elements. If you want to refer to an element in the matrix e.g. 62. In this, you would refer to the second element in the 1st row and 2nd column.

This is the way of presenting the Matrix rank. Thus, a rank of a matrix is defined as the maximum number of independent rows and independent columns.

Definition of Matrix Rank

The rank of a matrix is defined as the number of dimensions in any vector space, which is associated with the matrix. Here, by association, we mean that the rows of the matrix can be thought of as members of a vector space.

The dimension of that vector space is the rank of the matrix.

Another definition of Matrix Rank would be as follows :

The Matrix Rank is described as the maximum number of linearly independent column vectors or the maximum number of linearly independent row vectors in the matrix. Both of these definitions are equal.

For example, consider the r x c matrix, if r is less than c, then the maximum rank of the matrix is r and on the reciprocal, if r is greater than c, then the maximum rank of the matrix is c. Here, the rank of a matrix would be 0 only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.

Rank of a matrix and its properties

Here is another way to understand the rank of a matrix and its properties. Let’s consider a matrix of a p×qp×q matrix MM. Let rr be the smaller of the two numbers pp and qq.

Consider every possible r×rr×r submatrix of MM. If MM is square, you’ll have only one largest square submatrix: itself. If not, then you’ll have |p−q 1||p−q 1| such submatrices. If any one of these submatrices has a nonzero determinant, then the rank of MM is rr.

Otherwise, if all of these submatrices have a zero determinant, go to the next step.

If you’ve reached this step, you now know that the rank of your matrix is less than rr. Now consider every (r−1)×(r−1)(r−1)×(r−1) submatrix of MM. If any one of these submatrices has a nonzero determinant, then the rank of MM is r−1r−1. Otherwise, if all of these submatrices have a zero determinant, go to the next step.

If you’ve reached this step, you now know that the rank of your matrix is less than r−1r−1. Now consider every (r−2)×(r−2)(r−2)×(r−2) submatrix of MM. If any one of these submatrices has a nonzero determinant, then the rank of MM is r−2r−2.

On the other hand,  if all of these submatrices have a zero determinant, go to the further step. Repeat until you find a rank for your matrix, or until all of your 1×11×1 submatrices of MM have a zero determinant, in which case, the rank of your matrix is zero.

What has just been described above is called the determinantal rank, which can be proved to be equal to the rank of a matrix. Viewed in this way, the rank of a matrix is the size of its largest square submatrix whose determinant is nonzero.

Clearly, this method is terrible when it comes to computing the rank of a matrix.

Checking Matrix Equality

In Order to understand the equality of Matrix, you need to look into the below details. Any two matrices are equal in case all of the following conditions are met

1. The matrix should have the same number of rows
2. It should have the same number of columns.
3. The elements of each matrix should be equal

Consider an example of three matrices as stated below:

A =          111    x

y    444

B =        111    222

333    444

C =     l      m      n

o      p      q

Now, here is we want to consider A=B, we would state that x = 222 and y = 333. This is because the corresponding matrix is equal. We also know by the above example that C is not equal to A and B as C has more columns as compared to A & B.

Types of Rank Matrix

We also have three types of matrices such as Square matrix, transpose matrix & vectors.

The square matrix is mostly in a square shape. The numbers are arranged in squares in the same number of rows and columns. E.g.

1    2

2    3

Transpose Matrix is a matrix in which the transpose of one matrix is another matrix. This is found by using rows of the 1st matrix as columns of the 2nd matrix. E.g.

111    222

333    444

555    666

Next is Vectors which is only limited to either one row or either one column. E.g.

Row                                                Column

11                                             11     12    33

12

33′

Conclusion:

In this blog, we have explained a deeper version of the understanding matrix. By now you understand what is a matrix. It is a collection or set of vectors that are arranged in rows and columns.

You have also understood different types of matrices which are square, transpose and vector matrix. You also can go through the examples which are provided in the details to understand the matrix.