What is matrix reversible, the nature of matrix reversibility

The method of proving the reversibility of the matrix is as follows.

1. The rank of the matrix.

less than n, then this matrix is irreversible, and vice versa;

2. Matrix determinant.

If the value of is 0, then the matrix is irreversible, and vice versa;

3. For the homogeneous linear equation ax=0, if the equation has only zero solution, then the matrix is reversible, and if there is an infinite solution, the matrix is irreversible;

4. For the non-homogeneous linear equation ax=b, if the equation has only a special solution, then the matrix is invertible, and conversely, if there is an infinite solution, the matrix is irreversible.

1. Inverse matrix.

Let a be a field of numbers.

If there is another nth-order matrix b on the same number field, such that: ab=ba=e. Then we call b the inverse matrix of a, and a is called the invertible matrix.

Note: e is the identity matrix.

II. Definitions. An n-order square matrix A is called reversible, or non-singular, and if an n-order square matrix b exists, such that ab=ba=e

and say that b is an inverse matrix of a. A matrix that is irreversible is called a non-singular matrix.

The inverse matrix of a is denoted as a-1.

3. Nature. 1. The invertible matrix must be a phalanx.

2. (Uniqueness) If matrix a is reversible, its inverse matrix is unique.

3. The inverse matrix of the inverse matrix of a is still a. Write as (a-1)-1=a.

4. Transpose the matrix of the invertible matrix a.

At can also be reversed, and (at)-1=(a-1)t (the inverse of the transposition is equal to the inverse transpose).

5. If matrix a is reversible, then matrix a satisfies the elimination law. i.e. ab=o (or ba=o), then b=o, ab=ac (or ba=ca), then b=c.

6. The product of the two invertible matrices is still invertible.

7. The matrix is reversible if and only if it is a full-rank matrix.

4. Proof. 1. The inverse matrix is defined by the opposing matrix, so the inverse matrix must be a square matrix.

Let b and c be the inverse matrix of a, then there is b=c.

2. Suppose that b and c are both inverse matrices of a, b=bi=b(ac)=(ba)c=ic=ic, so any two inverse matrices of a matrix are equal.

3. By the uniqueness of the inverse matrix, the inverse matrix of a-1 can be written as (a-1)-1 and a, and therefore equal.

4. The matrix a is reversible, and there is aa-1=i. （a-1） tat=（aa-1）t=it=i ，at（a-1）t=（a-1a）t=it=i

5. From the definition of the invertible matrix, it can be seen that at is reversible, and its inverse matrix is (a-1)t. And (at)-1 is also the inverse matrix of at, by the uniqueness of the inverse matrix, so (at)-1=(a-1)t.

1) Multiply a-1 at both ends of ab=o at the same time (ba=o can also be proven), and get a-1(ab)=a-1o=o

And b=ib=(aa-1)b=a-1(ab), so b=o

2) It is proved by ab=ac (ba=ca), ab-ac=a(b-c)=o, and the two sides of the equation are the same as the left multiplication a-1, because a is reversible aa-1=i.

b-c=o, i.e., b=c.

The matrix a is a square matrix of order n, and if there is a matrix of order n such that the product of matrices a and b is a unit matrix, then a is called an invertible matrix and b is the inverse matrix of a. If the inverse of the square matrix exists, it is called an invertible matrix or a non-singular matrix, and its inverse matrix is unique.

Chinese name. Invertible matrix.

Foreign name. Invertible Matrix alias. Non-singular matrices.

Fast. Navigation.

Quality. Common methods.

Definition. Let be a field of numbers, and if there exists, such that , is a unit matrix, then it is called an invertible matrix, and the inverse matrix of is denoted as. If the inverse of the square matrix exists, it is called an invertible matrix or a non-singular matrix. [1]

Quality. 1) If it is an invertible matrix, then the inverse matrix is unique.

2) Let be an order matrix on a field of numbers.

If it is reversible, then and can also be reversed, and,;

If it is reversible, it is reversible, and;

Reversible. [1]

Common methods. 1) Common methods for judging or proving reversibility:

Prove; Find a matrix of the same order and verify;

The row vector (or column vector) of the proof is linearly independent.

2) How to find:

Formula method: where is the adjoint matrix of the matrix.

Elementary transformation method: When the pair is transformed into a unit matrix, the identity matrix is turned into.

N order phalanx a, if there is an n

The order square matrix b is such that ab=ba=in (or ab=in, ba=in, whichever satisfies one), where in

If it is an n-order identity matrix, it is called a

is reversible, and b

It is the a-counter-formation, which is written.

a^(-1)。

If the reverse of phalanx A exists, it is called A

It is a non-singular phalanx or a reversible phalanx.

The answer is a, at least one is irreversible.

b option, such as a

b is reversible, then ab is reversible;

Option C, AB can be irreversible, then AB is also irreversible;

Option d, a

One of the irreversible in b guarantees that ab is irreversible.

Properties of the Inverse Matrix:1. The invertible matrix is a phalanx.

2. Matrix A is reversible, and its inverse matrix is unique.

3. The inverse matrix of the inverse matrix of a is still a.

4. The transpose matrix at of the invertible matrix a is reversible, and (at)-1=(a-1)t.

5. If matrix a is reversible, then matrix a satisfies the elimination law.

6. The product of the two invertible matrices is still reversible.

7. The matrix is reversible only if it is a full-rank matrix.

Let a be an nth-order matrix on a number field, and if there is another nth-order matrix b on the same number field, such that: ab=ba=e, then we call b the inverse matrix of a, and a is called an invertible matrix. Note: e is the identity matrix.

Invertible matrix. The difference between the irreversible matrix and the irreversible matrix: the meaning is different, the representation is different.

The meaning of matrix A is reversible, which means that there is a matrix B, such that ab=ba=identity matrix.

A is known as an invertible matrix, and b is the inverse matrix of a.

Second, the representation is different: this proposition is a false proposition, for example, it can be overturned, for example, e and -e are both invertible matrices, but e+(-e)=o, the zero matrix is irreversible, so the proposition is false. The examples of an irreversible matrix multiplied by an invertible matrix as a zero matrix are only zero-moment rounds or matrices.

Matrix. It is a common tool in advanced algebra, and is also commonly found in applied mathematics disciplines such as statistical analysis, and in physics, matrices have applications in electrical circuits, mechanics, optics, and quantum physics. Computer science.

, the animation production of Santong Min Wu Wei also needs to use the matrix. The operation of the matrix is numerical analysis.

important questions in the field. Decomposing matrices into combinations of simple matrices can simplify the operation of matrices in theory and practical applications.

Sufficient and necessary strips of matrix reversible: ab=e; a is a full-rank matrix (i.e., r(a)=n); The eigenvalues of a are not 0 at all; The determinant of a |a|≠ round withering 0, it can also be expressed as a non-singular matrix (i.e., a matrix with a determinant of 0).

a is equivalent to an nth-order identity matrix; a can be expressed as the product of the elementary matrix; The system of homogeneous linear equations ax=0 has only zero solutions; The system of nonhomogeneous linear equations ax=b has a unique solution; The group of row (column) vectors of a is linearly independent; Any n-dimensional vector can be linearly represented by a set of row (column) vectors of a.

(a^-1)(a+b)(b^-1)=b^-1+a^-1

Since the inverse of the reversible array is reversible, the product of the reversible array is reversible, which is known from the above formula: a -1

b-1 reversible. Then by the return car nature: (ab) -1 = (b -1) (a -1) by (** formula, both ends of the inverse.

Obtain: (a -1+b -1) -1==[b -1)] 1}[(a+b) -1][(a -1) -1]=(b)[(a+b) -1](a).

Properties of invertible matrices:

1. The invertible matrix must be a phalanx.

2. If matrix a is reversible, its inverse matrix is the only one that can be discussed.

3. The inverse matrix of the inverse matrix of a is still a. Write as (a-1)-1=a.

4. The transpose matrix at of the invertible matrix a is also invertible, and (at)-1=(a-1)t (the inverse of transposition is equal to the inverse transpose).

5. If matrix a is reversible, then matrix a satisfies the elimination law. That is, ab=o (or ba=o), then b=o, ab=ac (or ba=ca) is resistant to the world, then b=c.

There is only a bai phalanx.

It is possible to be reversible, and it is impossible to talk about him without the du matrix of the phalanx.

Zhi reverse. If there is an nth-order matrix b, such that the product of the version matrix a and the weight b is a unit matrix, then a is called an invertible matrix, and b is the inverse matrix of a. If the inverse of the square matrix exists, it is called an invertible matrix or a non-singular matrix, and its inverse matrix is unique.

No. 1. The primary matrix must be reversible.

2. Matrix. The number table of m rows n columns arranged by m n number aij is called m row n column matrix, referred to as m n matrix. Shaped like the answer

The number of m n is called the element of matrix a, which is referred to as the element, the number aij is located in the ith row of matrix a, column j, which is called the (i,j) element of matrix a, and the matrix with the number aij as the (i,j) element can be denoted as (aij) or (aij) m n, and the m n matrix a is also denoted as amn.

3. Elementary matrix:

An elementary matrix is a matrix obtained by an identity matrix through three elementary transformations of the matrix.

4. Reversible: Let a be an nth-order square matrix on the number field, and if there is another n-order matrix b on the same number field, such that: ab=ba=e. Then we call b the inverse matrix of a, and a is called the invertible matrix. e is the identity matrix.

5. Calculation method:

Verify that the two matrices are inverse matrices to each other:

According to the multiplication of the matrix, ab=ba=e, so a and b are inverse matrices of each other.

Proof: If matrix A is reversible, then the inverse matrix of A is unique.

If b and c are both inverse matrices of a, then there is:

So b=c, i.e., the inverse matrix of a is unique.

Elementary transformation method of inverse matrices:

No. First of all, only the phalanx can be reversed, and it is impossible to talk about his inversion without the matrix of the phalanx.

Second, even the square matrix is not necessarily reversible, because one of the sufficient and necessary conditions for the reversibility of a matrix is that its determinant is not 0, and when the determinant of the matrix is equal to 0, the matrix must be irreversible.

It is that all matrices can be converted into standard type, and the standard type here refers to the equivalent standard type of the matrix.

Let the rank of the matrix a be r(a)=r, then a must be reduced to the equivalent standard type er o

o o Hope it helps.