Elementary transformations of matrices and elementary matrices

In this question, first look at the rank of A and B, because Pa=B, Rank(B)=min(rank(p),Rank(A)), so if the Rank of B is less than the rank of A, the rank of P must be less than the rank of A, then P is irreversible and there is no solution. But here rank(a)=rank(b)=2, so the rank of p must be greater than or equal to 2, and it is reversible when the rank of p is equal to 3. As for how to solve p, my solution is to convert pa=b to a'p'=b', and then solve p first', the method used is Gaussian elimination to put p'The vector space of each column is solved and then converted to p.

Using the computer to calculate the results of p=, , which is the solution of all vector spaces, r, s, t are arbitrary real numbers. In this case, p can be eliminated by the Gaussian elimination method'To test the rank of p, the eliminated p' becomes,} that is, the rank of p is only related to t, and when t is not 0, the rank of p is equal to 3 and is reversible. So the general solution for p is that ,r,s is any real number and t is not 0.

By the way, help you calculate p -1=, ,

Linear algebra. I'll tell you tomorrow, I'm sorry I don't have a pen or paper right now

Matrix factorization is the decomposition of a matrix into the sum or product of a relatively simple or several matrices with certain characteristics, and the matrix factorization methods generally include triangular factorization, spectral factorization, singular value factorization, full-rank factorization, etc.

The problem is not described, what do you mean by not ending it?

What does it mean to perform the primary transformation of "Xingxing"?

For a matrix, row transformations and column transformations can be done.

Sometimes there is a difference, and the key depends on what your purpose is.

For example, when it is used to seek adversity, it can only be used as rows or columns; Solve a system of linear equations, and only make row transformations for augmented matrices or coefficient matrices; It is used to find the rank of the matrix, and the column and column transformation can be done as you want.

Row transformation, column transformation, are all done step by step, what do you mean by beauty before the end?

Three categories: two rows (columns) of the fabric

A row (column) of a matrix is multiplied by a non-zero number.

A row (column) of a matrix multiplied by a non-zero number added to another row (column) of the matrix does not change the rank of the matrix.

The rank does not change after the matrix is transposed.

The definition of the elementary transformation of the broad array is the elementary row transformation and the elementary column transformation of the matrix, which is an important calculation tool in linear algebra, a noun in advanced algebra, and the name of an operation.

1. Types of matrix elementary transformations

2. Multiply all the elements of a row of a matrix by a non-zero number k (multiply k by k in line i).

3. Multiply all the elements in a row of the matrix by a number k and add them to the corresponding elements in another row (multiply the j line by k and add to the ith line as ri+krj).

4. Similarly, changing the above "row" to "column" will give the definition of the matrix elementary transformation, and replace the corresponding symbol "r" with "c".

Second, the rules of matrix transformation.

1. Wrap transformation: Swap two rows (columns), that is, ri rj (or ci cj for columns).

2. Multiplier transformation: multiply all the elements of a row (column) of the field light of the determinant by the number k, that is, ri k(k≠0) or ri k(k≠0).

3. Elimination transformation: multiply all the elements of a row (column) of the determinant by a number k and add them to the corresponding elements of another row (column), that is, ri+rj k or ri+rj k.

1. The primary matrix refers to the matrix obtained by the identity matrix through an elementary transformation, 2. There are three kinds of primary transformations (for one.

Multiply the number of rows or columns (not a number of 0s), add the multiple of one row (column) to another row (column), and exchange two rows (columns). For each elementary transformation, there is an elementary matrix.

3. The original matrix is multiplied by an elementary matrix on the left, which corresponds to an elementary row transformation of the original matrix by a primary matrix on the right, which corresponds to an elementary column transformation.

Elementary transformations of matrices refer to some simple transformations of matrices by basic operations on matrices, including:

two rows or two columns of the fabric;

Use a non-zero multiplication matrix of a row or column of a noisy pants;

Multiply a row or column of a matrix by a non-zero number and add it to another row or column.

These transformations can be expressed by matrix multiplication, i.e., left multiplication by an elementary matrix:

Two rows or two columns of the fabric: Let the i-row and j-s (or columns i and j) of the fabric be interchanged, then the corresponding elementary matrix is eij, that is, the main diagonal elements of the matrix are all 1, except for the elements of row i and j-(or columns i and j) that are 0, the elements of rows i and j, or columns i and j, are 0 and 1, respectively, i.e.

eij = 1]

Multiply a row or column of a matrix with a non-zero number: Let the i-row (or i-column) of the matrix be multiplied by the non-zero number k, then the corresponding elementary matrix is ei(k), i.e., the main diagonal elements of the matrix are all 1, except that the elements in row i-i (or column i) are k, i.e.

ei(k) =1]

0 0 ..k 0]

Multiply a row or column of a matrix by a non-zero number and add it to another row or column: Let the j-row of the matrix be multiplied by the non-zero number k and then added to the i-row (or multiplied by the j-column of the matrix by the non-zero number k and added to the i-column), then the corresponding elementary matrix is eij(k), that is, the main diagonal elements of the matrix are all 1, except for the elements in row i and j, (or columns i and j) are 0, and the i-th element of row j (or column j) is k, i.e., .

eij(k) =1]

0 0 ..k ..0]

Through elementary transformation, matrices can be transformed into row step matrices or minimal matrices, so that it is convenient to solve problems such as linear equations or rank of matrices.

A matrix elementary transformation is a transformation that uses a matrix of left or right multiplication. Specifically, matrix elementary transformations include three basic transformations: swapping two rows, swapping two columns, and multiplying elements of a row or column. Below I will explain these three basic transformations in detail.

1.Swap Two Rows: Swap the positions of two rows in the matrix. For example, for a 3x3 matrix, swap rows 1 and 2 to get a new matrix:

begina_ &a_ &a_ \

a_ &a_ &a_ \

a_ &a_ &a_

end$2.Swap Two Columns: Swaps the positions of two columns in the matrix. For example, for a 3x3 matrix, Na Saura swaps columns 1 and 2 to get a new matrix:

begina_ &a_ &a_ \

a_ &a_ &a_ \

a_ &a_ &a_

end$3.Multiply the elements of a row or column: Multiply the elements in a row or column by a non-zero constant k. For example, for a 3x3 mega matrix, you can multiply all elemental family answers in row 2 by 2 to get the new matrix:

begina_ &a_ &a_ \

2a_ &2a_ &2a_ \

a_ &a_ &a_

End$In general, the elementary transformation of matrices is a key step in the determinant calculation and solution of linear equations. In matrix operation, the elementary transformation of the matrix will not change the rank and matrix equivalence relationship of the matrix, but it can easily solve the problems of the determinant, inverse matrix, and linear equation system of the matrix.

Let the two squares a(n*n) and b(m*m) be on the subdiagonal, and move a,b to the main diagonal by the column transformation of the matrix, and then use Laplace. The first column of a is transformed m times, the second column of a is also m times, and so on, the column transformation of the nth column of a is also m times, and we can know that the column transformation is m * n times, and after the column transformation is completed, b has moved to the main diagonal, so it is necessary to multiply (-1) (m*n).

Let the two squares a(n*n) and b(m*m) be on the sub-diagonal, and move a,b to the main diagonal through the column transformation of the moment to bury the array, and then use Laplace. The first column of a is transformed m times, the second column of a is also m times, and so on, the column transformation of the nth column of a is also m times, and we can know that the column transformation is m * n times, and after the column transformation is completed, b has moved to the main diagonal, so it is necessary to multiply (-1) (m*n).

Proper partitioning of matrices can make the operation of higher-order matrices can be transformed into the operation of low-order matrices, and at the same time, the structure of the original matrix can be simple and clear, so that the operation steps can be greatly simplified, or the theoretical derivation of the moment is convenient.

Elementary algebra starts with the simplest unitary equations, and on the one hand, it goes on to discuss binary and ternary systems of one-dimensional equations, and on the other hand, it studies systems of equations that are more than two-dimensional and can be converted into two-dimensional. Continuing in these two directions, algebra discusses a system of one-dimensional equations with any number of unknowns, and also refers to a system of linear equations while also studying a system of unary equations of a higher degree.

At this stage, it is called advanced algebra. Advanced algebra is a general term for the development of algebra to an advanced stage, and it includes many branches. The advanced algebra offered in universities now generally includes two parts: linear algebra and polynomial algebra.

There are 3 situations in which the primary row (column) of the matrix is transformed:

1. A row (column), multiplied by a non-zero multiple.

2. A row (column), multiplied by a non-zero multiple, added to another row (column).

3. Two rows (columns) are interchanged.

It is easy to see that none of these three elementary transformations change the non-zero nature of the determinant of a square matrix, so if a matrix is a square matrix, we can judge whether the original matrix is reversible by looking at whether the matrix after the elementary transformation is reversible.

It can be seen that the three elementary transformations of the matrix are all reversible, and their inverse transformations are also the same type of elementary transformations.

This is something I learned a long time ago.,I vaguely remember this transformation.,But I don't remember the details very well.,Now it seems that I've learned a lot.。 Review the old and learn the new.

The elementary transformations of matrices are divided into elementary row transformations and elementary column transformations, and the types of column transformations and row transformations are similar, so only the elementary row transformations are mentioned here.

Typically, we write the elementary row transformation above the arrow and the elementary column transformation below the matrix. The new matrix obtained by a finite order elementary transformation is equivalent to the original matrix. MATLAB uses the rref() function to calculate the simplified trapezoidal form of the matrix.

Elementary transformations can be expressed by elementary matrices, which are matrices obtained from the identity matrix through an elementary transformation.

For the elementary row transformation, the elementary matrix is left multiplied by the original matrix; For elementary column transformations, multiply the elementary matrix by the original matrix.

There are a few things we can do with it:

The above three operations are collectively referred to as the "matrix".Elementary row transformations

theInverse transformationAs follows:

If the object for which these operations are directed is a column of a matrix, it is called ".Elementary column transformations

A matrix with a finite order elementary transformation, and the original matrixEquivalent。If the finger has a row transformation, it is called ".Line equivalence”；Only column transformations were made, called ".Column equivalence

Passed by the identity matrix eOnceThe matrix obtained by the elementary transformation is calledElementary matrices

Performing an elementary transformation on a general matrix is equivalent to multiplying the matrix with an elementary matrix that has performed the same elementary transformation.

Let a be a m n matrix, such as a modulation transformation, let's take a look:

Swapping two rows is equivalent to multiplying the original matrix with an elementary matrix; Swapping two columns is equivalent to multiplying the primary matrix by the original matrix.

For other elementary transformations, the left-row-right-column rule is also followed. You can verify it yourself.

For n 2n order matrices (a|e) Implementation of the primaryOKtransformation, when a becomes e, e becomes the inverse matrix of a: