This paper discusses the invariance of the rank of a matrix, the conditions and applications of matrix inequalities, the relationship between matrix rank and matrix operations, the relationship with matrix reversibility, the linear correlation with vector groups and the zero relationship of eigenvalue algebra multiples and other properties. The application of the rank of the matrix in linear algebra, analytic geometry, probability theory, etc. is obtained.
Keywords
- matrix rank
- matrix rank invariance
- matrix rank inequality
- matrix rank identity
- linear equations
- zero eigenvalue algebraic multiples
- homogeneous linear equations
MSC 2010
- 15B99
The modern concept of the matrix was gradually formed in the 19th century. In 1801, German mathematician Gauss took all the coefficients of a linear transformation. In 1844, German mathematician Eisenstein discussed ‘transformation’ (matrix) and its product. In 1850, the British mathematician Sylvester first used the term matrix. In 1858, British mathematician Gloria published ‘Research Report on Matrix Theory’. He first studied matrices as an independent mathematical object and first published a series of articles on this subject. Therefore, he was considered the founder of matrix theory. He gave a series of definitions that are commonly used now, such as two matrices, equality, zero matrix, sum of two matrices, quantity product of a number and a matrix, product of two matrices, matrix inverse, transposed matrix, etc [1]. Kelley also noticed that matrix multiplications are combinable but generally not commutative and
The maximum order
The ranks of the transposed matrices are equal, that is,
Suppose that the
Suppose that row
Let's multiply the If If If Since
From the above three cases, we can see that all the sub-forms of matrix
In fact, performing a row or column elementary transformation is equivalent to multiplying this matrix to the left or right by an invertible matrix. Let
We only need to further subtract the appropriate multiples of line
In fact, the matrix is transformed into a step shape by elementary transformation, and the rank of the number of non-zero rows in the step shape matrix is the rank of the matrix. This results in another equivalent definition of matrix rank:
The number of non-zero rows in the staircase formed by matrix
The elementary transformation is used to transform the matrix
Any matrix
If the matrix
If
Set
If the rank of matrix
The matrix of contracts has the same rank. That is, if
If
0 ≤
Let
So, have:
This property is extended to the case of the product of any
First prove
(Sylvester inequality) Let
(Proven using a block matrix) Since
If the matrices
Prove: because
Because
If
Because
Among them,
If one of
This results in the conditions for the equality of Sylvester and Frobenius inequality:
Assume
If (
With Theorem 1.4.4
Assume
If
If
Assume
due to (
Assume
due to (
Analysis of typical examples:
prove:
w make
The above three examples are very cumbersome to solve with knowledge of zeroization polynomials, but it is very simple and easy to understand using the Sylvester inequality. Matrix inequality has good applications in solving problems, and this article will not explain them one by one.
For any matrix
Let
Use the elementary transformation indicated by Lemma 1.1.1 above to turn
The rank of a matrix is a basic concept and one of the most important quantitative characteristics of a matrix. It is an invariant under elementary transformation. The rank of a matrix is an important concept that reflects the inherent characteristics of a matrix. It has an important role in linear algebra, analytic geometry and even probability theory.
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