What are singular and non singular matrices

1) A singular matrix is a linear algebraic concept, that is, the rank of the matrix is not a full rank. First, see if the matrix is a square matrix (i.e., a matrix with an equal number of rows and columns. If the number of rows and columns is not equal, then there is no singular and non-singular matrices).

Then, let's look at the determinant |a|whether it is equal to 0, if it is equal to 0, the matrix A is called a singular matrix; If it is not equal to 0, the matrix a is called a non-singular matrix.

At the same time, by |a|≠0 shows that matrix a is reversible, so we can draw another important conclusion: the invertible matrix is a non-singular matrix, and the non-singular matrix is also an invertible matrix.

If a is a singular matrix, then ax=0 has an infinite solution, and ax=b has an infinite solution or no solution. If a is a nonsingular matrix, then ax=0 has and only a unique zero solution, and ax=b has a unique solution.

2) Non-singular matrices:

If the determinant of the nth order square matrix a is not zero, i.e.

a|≠0, a is called a non-singular matrix or a full-rank matrix, otherwise a is called a singular matrix or a descending matrix.

The singular matrix is the determinant for.

(it must be a square matrix to talk about singular and non-singular), that is, irreversible matrices, and non-singular matrices are determinant not.

is the invertible matrix.

A matrix with a determinant of 0 is a singular matrix, and a matrix that is not 0 is a non-singular matrix.

If the determinant of the nth order matrix a is not zero, i.e.

a|≠0, a is called a non-singular matrix, otherwise a is called a singular matrix.

A non-singular matrix is also known as a non-degenerate matrix, also known as a full-rank matrix, an important and widely used special matrix, and the nth order matrix a of the determinant A ≠0 on the number field p is called a non-singular matrix, if |a|=0, then a is called the singular matrix, also known as the degradation matrix.

Nonsingular matricesAnother matrix is an important tool used to describe the scattering experiments that form the cornerstone of experimental particle physics. When particles collide in the accelerator, the particles that have not interacted with each other enter the action zone of other particles in high-speed motion, and the momentum changes, forming a series of new particles.

This collision can be interpreted as a scalar product of the linear combination of the resulting particle state and the incident particle state. The linear combination can be expressed as a matrix, called an s-matrix, in which all possible interparticle interactions are recorded.

Linear transformation and symmetry of non-singular matrices:

Linear transformations and their corresponding symmetry play an important role in modern physics. For example, in quantum field theory, elementary particles are represented by Lorentz groups of special relativity, specifically, their behavior under spinor groups. The specific representation of Pauli matrices and more generally, Dirac matrices, is an indispensable part of the physical description of fermions.

Whereas, the performance of fermions can be expressed in terms of spinors. To describe the lightest three quarks, it is necessary to use a group theory representation with a special unitary group su(3); Physicists use a simpler matrix called the Gell-Mann matrix when they calculate, which is also used as the su(3) gauge group, and the modern description of strong nuclear forces quantum chromodynamics is based on su(3).

There is also the Kabibo-Kobayashi-Ikawa matrix (CKM matrix): the fundamental quark states that are important in weak interactions are not the same as those with different masses between specified particles, but the relationship between the two is linear, and this is what the CKM matrix represents.

A singular matrix is a concept of linear algebra, that is, a matrix whose corresponding determinant is equal to 0.

For a non-zero matrix A with n rows and n columns, if there is a matrix b such that ab=ba=i (i is the identity matrix), then a is said to be reversible, and a is also called a nonsingular matrix. The implicit in this definition is that the singular matrix is a phalanx because the determinant is in relation to the square matrix. A determinant of exactly zero is "singular".

Reasons why the singular matrix is singular:

The coefficient determinant may take various values, but whatever the value is, as long as it is not zero, the solution of the corresponding system of equations must be unique. However, if the coefficient determinant happens to be zero, there can be an infinite number of solutions to the system of equations.

In this way, a matrix with a determinant of zero appears to be "prominent", very "different", very "alternative", very "strange", and so on. And "singular" encompasses both strange and heretical, and is used to describe this kind of matrix.

The meaning of the singular matrix is that the rank of the matrix is not full rank.

A singular matrix is a linear algebraic concept, i.e., a square matrix where the corresponding determinant is equal to 0.

How to judge the singular matrix:

First of all, see if the matrix is a square matrix (i.e., a matrix with an equal number of rows and columns, if the number of rows and columns is not equal, then there is no singular matrix and no singular matrix).

Then, let's look at the determinant |a|whether it is equal to 0, if it is equal to 0, the matrix A is called a singular matrix; If it is not equal to 0, the matrix a is called a non-singular matrix.

At the same time, by |a|≠0 shows that matrix a is reversible, so we can draw another important conclusion: the invertible matrix is a non-singular matrix, and the non-singular matrix is also an invertible matrix.

If a is a singular matrix, then ax=0 has an infinite solution, and ax=b has an infinite solution or no solution. If a is a nonsingular matrix, then ax=0 has and only a unique zero solution, and ax=b has a unique solution.

The Singular Matrix isIrreversibleof the matrix. As we all know, matrices describe linear transformations. If this transformation is reversible, it is regular; The opposite is "".Strange (singular)."of.

For example: (90° clockwise), its inverse is (90° counterclockwise).

Another example: if a multidimensional space is compressed to a point (i.e., a 0 matrix), the transformation is irreversible. Because of youIt is not possible to reverse expand a point into a space

If it is reversible, should the transformed original multi-dimensional space be one-dimensional or two-dimensional and three-dimensional? Or maybe even a two-dimensional plane in three-dimensional space?

This spatial compression is due to the basis vectors that represent the transformationsLinear correlation, or ratherDeterminant(Ratio per unit of space) = 0.

WhyNopeReversibleYesStrange"of. It can be understood like this:

Linear transformations are made up of severalBasis vectorsto represent.

VectorsLinearity is not relatedNormalityRelevance is special. For example, in two-dimensional space, two vectors are obviously not collinear than collinear. The same goes for high dimensions.

Linearity independence means that there is no dimensionality reduction and it is reversible. ThereforeReversibility is the norm, irreversible is "singular".

There is also an angle, for ax=b,Singularity means that there may be no solution

Linear transformations are made up of severalBasis vectorsto represent.

For example, in a two-dimensional space, two vectors that are not collinear can be combined to form all vectors; But once collinear, there may be no solution (singular).

The name of singular matrix comes from the English singular matrix, I think the main reason is that singularity will occur when seeking inverse, the matrix is actually a linear map, and the singular matrix corresponds to an irreversible map.

In addition, if the elements of the matrix are all real numbers (or complex numbers) and satisfy a certain continuous distribution, then the probability that its determinant is zero is zero, in this sense the singular matrix itself is indeed a very strange matrix, but this is only from the Chinese point of view, there is no such meaning in the English name.

Singular Matrix. It refers to the blue style of the ranks.

is 0 for the matrix.

The following three points are equivalent to the type quietly:

1. A is the singular matrix.

2. The determinant of a is 0

3. A is irreversible.

Also known as non-degenerate matrix, also known as full-rank matrix, an important and widely used special matrix, the nth order matrix a of the determinant a ≠0 on the number field p is called a non-singular matrix, if |a|=0, then a is called the singular matrix, also known as the degradation matrix, also known as the descending matrix. The matrix a is non-singular if and only if a is reversible or a can be expressed as the product of several elementary matrices.