What is the rank of a matrix and what is the rank of a matrix?

In layman's terms, if you think of a matrix as a row vector or column vector, the rank is the rank of these row vectors or column vectors, that is, the number of vectors contained in a greatly independent group.

The column rank and row rank of a matrix are always equal, so they can simply be called the rank of matrix a. It is usually expressed as R(A), Rk(A), or Rank A.

Using the elementary row transformation, it is formed into an upper triangular array.

In a stepped shape, there are as many non-zeros as there are ranks.

The rank of the determinant is as follows:For a determinant, the highest order of a non-zero sub is its rank. The rank of the matrix.

It is used to represent a matrix structure.

Indicates whether some rows of the matrix can be replaced by others.

, the column rank of a matrix a is linearly independent of a.

The largest number of tandems. Similarly, the row rank is a linear independent maximum number of rows.

Features of the determinant:A row in the determinant A is multiplied by the same number k, and the result is equal to ka.

The determinant a is equal to its transposed determinant at (the ith line of at is column ith of a). Noisy only.

If the nth order determinant |αij|In a row (or column) is promoted, the determinant is |αij|is the sum of two determinants, the ith row (or column) of these two determinants, one is b1, b2 ,...,bn；The other is 1, 2,...,n；The meta on the remaining rows (or columns) is the same as |αij|of exactly the same as accompanying Tan.

First of all, =(a1,a2,a3,an) t is a column vector. And every element in the vector is not 0, so the rank of aat is equal to 1 (the rank of a single vector cannot be greater than 1).

In the same way, t is a row vector, so the rank of t is also equal to 1.

a=ααt。

According to the nature of the matrix rank.

The smaller number of the rank of ab and the rank of b.

So the rank of a is a rank and the rank of t is a smaller number.

i.e. rank 1 of a.

At the same time, because and t are not 0 for each element.

So every element of the A matrix is also not 0, so the rank of a cannot be 0, so the rank of a is 1.

The rank of the matrix.

Theorem: The row rank, column rank, and rank of a matrix are all equal.

Theorem: Elementary transformations do not change the rank of the matrix.

Theorem: If a is reversible, then r(ab) = r(b) and r(ba) = r(b).

Theorem: the rank rab<=min of the product of the moment seepage array;

Lemma: Let the column cluster of matrix a=(aij)sxn be equal to the number of columns n of a, then the rank of the column of a is equal to n.

When r(a)<=n-2, the order of the highest order non-zero sub-formula <=n-2, and any n-1 sub-formula is zero, and the elements in the adjoining matrix are n-1 sub-formulas plus a plus or minus sign, and the adjoint matrix is returned to the 0 matrix.

The above content refers to: Encyclopedia - Rank of the matrix.

Here's why:

Let a be a matrix of m n, which can be proved by ax=0 and a'ax=0 two n-ary homogeneous equations.

r(a'a)=r(a)。

1. ax=0 is definitely a'ax=0, easy to understand.

2、a'ax=0 → x'a'ax=0 → ax)' ax=0 →ax=0。

Therefore the two equations are solved in the same way.

The same can be said for r(aa')=r(a')。

In addition there is r(a)=r(a')。

So in summary, r(a)=r(a')=r(aa')=r(a'a)。

Rank inequality of matrices.

1) The rank of matrix a is equal to the transposed rank of matrix a, that is, the row rank of the matrix = column rank.

Proof-of-the-art idea: A matrix undergoes a series of elementary transformations.

can all correspond to a standard type, and the number of non-zero rows of the standard type is the rank of the matrix. And because the standard type of the matrix is unique, the row rank of the matrix must be equal to the column rank of the matrix.

2) The rank of matrix a is equal to the rank of matrix a transposed by matrix a.

Proof idea: construct a system of linear equations of homogeneous order respectively.

ax=0 is the same solution as a transposed by ax=0. Because the previous equation can be used.

The sub pushes to the next equation, and vice versa, and the reverse also holds. The two systems of equations are solved in the same way, so the rank is equal, and the code match is proved.

The number of vectors in a maximally linear-independent group of row vectors or column vector groups of a matrix.

First of all, the key is supposed to be a homogeneous system of linear equations. The number of equations is less than the number of unknowns, that is, the rank of the coefficient matrix is less than the number of unknowns.

I think it's easier to understand that the rank of the coefficient matrix is the number of valid equations.

The number of unknowns is more than the number of valid equations, and naturally there are non-zero solutions.

Similar to x+y=3, an equation with two unknowns, x y, naturally has a non-zero solution.

Important theorems. Every linear space has a base.

For a non-zero matrix A with n rows and n columns, if there is a matrix b such that ab = ba = e (e is the identity matrix), then a is a nonsingular matrix (or invertible matrix) and b is the inverse matrix of a.

A matrix is nonsingular (reversible) if and only if its determinant is not zero.

A matrix is non-singular if and only if the linear transformation it represents is automorphic.

A matrix semi-definite carries sails if and only if each of its eigenvalues is greater than or equal to zero.

The matrix is positive definite only if each of its eigenvalues is greater than zero.

The above content refers to: Encyclopedia - Linear Algebra.

Simply put, it is a vector number with a solution.

For example, if you say more: rank is the number of non-0 rows of a stepped matrix, why?

Because if it is line 0 (after the transformation of the elementary row), 0x1 + 0x2 + 0x3 + 0x4 + 0x5 + ......0, which does not help to solve this equation, cannot be included in the rank. (x is an unknown number, not a multiplier sign).

Again, why the nanobeat rank is a maximally linear independent group.

1x1+2x2+3x3=0

2x1+4x2+6x3=0

You will notice that the two equations are actually the same, which is called linear correlation.

We can also do this with elementary row transformations.

r2-r1 times 2 = 0, rank 1

From a spatial point of view, rank is the dimensionality occupied by the matrix, for example, we can solve three unknowns with a system of ternary equations, (three unknowns of the three repentant equations).

Then we call it full rank.

It can be understood as three unknowns, namely the x-axis, y-axis, and z-axis, which can form a three-dimensional space.

But if there is a useless solution, in fact, it is no longer three equations, then it is not ranked, and then the basic solution system will be introduced.

The above only discusses homogeneous linear equations.

And it's not accurate, it's only for beginners.

Question 1: What is the rank of the matrix After the positive volt matrix is transformed into elementary rows, the number of non-zero rows is called the row rank, and the number of non-zero columns is called the column rank.

The vertical sock rank of the matrix is the row rank or column rank after the elementary row transformation or column transformation of the square matrixProblem 2: What is the rank of the matrix Trouble to explain in layman's understanding 10 points Just the number of equations, how do you usually solve equations, do you just add and subtract two equations from each other, sometimes you add and subtract the equations and finally you will find that there are a pair or even more equations that are the same, these same equations are equivalent to one equation, and then add other messy equations, which is the rank.

Question 3: What exactly is a rank in linear algebra is a numerical feature of a matrix! He is an intrinsic property of a matrix! This is the largest number of rows or columns of a non-zero child!

Question 4: What does the rank property 4 of the matrix mean? You are talking about properties that if p and q are reversible, then r(paq)=r(a) is the representation here.

For a left-powered or right-powered invertible matrix of matrix a, its rank remains unchanged That is, the re-excitation elementary transformation of the matrix does not change the rank of the matrixQuestion 5: What does the rank of the matrix mean What do you mean if the property p and q are reversible, then r(paq)=r(a).

Here's the representation.

For a left-powered or right-powered invertible matrix of matrix a, its rank is unchanged, that is, the elementary transformation of the matrix does not change the rank of the matrix.