The development process of matrix simplification, and several methods of matrix simplification are i

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Here's how to simplify <> matrix:

1. Use elementary rigid transformation to simplify. Use the line transformation to make each line into the simplest form, that is, observe the numerical characteristics of each line, select the line that needs to be simplified, add a suitable multiple of a row, and turn it into the simplest form.

2. Then use the column transformation to zero the rest of the elements of the row where the first non-zero element of each non-zero row is located, making it the simplest form.

3. Appropriately swap the positions of each column so that the upper left corner becomes a unit array.

4. Identity matrix.

It is the simplest form of the matrix, and the transformation of a matrix into an identity matrix is the simplest form of splitting friends.

The elementary transformation matrix is mainly carried out in order, first into the row ladder, and then into the row minimalist form.

For example, it is easier to make the element in the first column of the first row 1 1, and then use this 1 to turn the element below 1 into zero;

In the same way, it is easier to make the elements in a certain row and column 1 1, and use this 1 to turn the elements below 1 into zero;

Also, turn fractions into integers first to avoid fractional operations;

Also, observe the relationship between the elements in the matrix, whether it is numbers or letters, and perform some tricky calculations.

The primary transformation matrix is used to make the line simplest shape, which is mainly carried out according to the order of combustion and closing illumination, first into the row ladder shape, and then into the line minimum shape. For example, it is easier to first make the first row of the first row of skin or crack elements 1, and use this 1 to turn the elements below 1 into zero; In the same way, after the source of the group, the elements in a certain row and column are 1, and it is relatively simple to use this 1 to turn the elements below 1 into zero; 3 transformations of elementary row transformations:

1. Take a non-zero in p'A row of the multiplication matrix.

2. Add the c times of one row of the matrix to another row, where c is any number in p.

3. Swap the positions of the two rows in the matrix.

Generally speaking, a matrix becomes another matrix after the elementary row transformation, and when matrix A becomes matrix B after the elementary row transformation, it is generally written as a b

It can be proved that any matrix can always become a stepped matrix after a series of elementary row transformations.

Hello, after diagonalizing the matrix, the matrix to the nth power is the nth power of each element in it.

Let the linear transformation a, the matrix under the basis m be a, the matrix under the basis n be b, and the transition matrix from m to n be x, then we can prove: b=x ax

Then the definition: a, b are 2 matrices. If there is an invertible matrix x, satisfying b=x ax, then a is said to be similar to b (an equivalence relation).

If there is an invertible matrix x that makes a similar to a diagonal matrix b, then a is said to be diagonalizable.

Correspondingly, if the matrix of the linear transformation A under the basis m is A, and A is similar to the diagonal matrix B, then let X be the transition matrix, i.e., the number of rocks can be found by finding the basis n, and the moment chaos array of A is linearly transformed into a diagonal matrix under n, thus achieving simplification.

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Here's how to simplify <> matrix:

1. Use elementary rigid transformation to simplify. Use line transformation to make each line into the simplest form, that is, observe the numerical characteristics of each line, select the line that needs to be simplified, add an appropriate multiple of a row, and turn it into the simplest form, and follow this step to make each line that needs to be simplified into the simplest form.

2. Qin Huiqi then uses column transformation to zero the rest of the elements of the row where the first non-zero element of each non-zero row is located, making it the simplest form.

3. Properly exchange the positions of each column, and it is too early for Biyou to make the upper left corner of a unit.

4. The identity matrix is the simplest form of the matrix, and the simplest form is transformed into an identity matrix.

There are many ways to simplify matrices to row minimalist matrices, generally using invertible matrices for deterministic transformations, and in numerical calculations, orthogonal transformations and triangle transformations are often used.

1. QR decomposition of the matrix: Q is an orthogonal matrix, and R is the upper triangular matrix. There are two ways to decompose a QR from a matrix.

One is the Gram-Schmidt orthogonalization method. The advantage of this method is that no matter how many steps are broken down, Chunbi can stop halfway. The modified Gram-Schmidt orthogonalization method obtained by this method can also be regarded as the Arnoldi method as a fast eigenvalue method for matrices.

For more information, see Knowledge about the Krynov subspace.

The second is the household orthogonal triangulation method, which essentially uses the mirror transformation operator to reduce the triangular part of the original matrix to 0. Finally, we can get an upper triangular matrix. The disadvantage of the method is that it cannot be stopped halfway.

2. SVD decomposition of matrices: An MXN matrix can be reduced to a unit matrix and a splicing of null matrices by multiplying the orthogonal matrix. SVD (Singular Value Decomposition), as the name suggests, is a type of decomposition that can be applied to any matrix.

It is widely used in solving low-rank matrix approximations.

3. Gauss elimination method. This is also a way to reduce the matrix to the standard type. Finally, you can get a matrix of trouser corners on the third hand. The purpose is to solve systems of linear equations. The advantage is that the calculation is simple, and the disadvantage is that the stability analysis is too complex.

4. SCHUR decomposition: a complex matrix is transformed into an upper triangular matrix by using the unitary similarity transformation. When the complex matrix is a Hermitian matrix, a diagonal matrix can be obtained at the end.

This is a simplification of the determinant of the matrix, and we know that the elementary transformation of rows and columns on the determinant does not change the value of the determinant, so we transform it as follows:

1. Multiply the first row of the determinant by -1 and add to the second and third rows respectively:

2. Add the third column of the determinant to the first column:

3. Add the second column of the determinant to the first column:

4. Multiply the second row of the determinant by the reciprocal and add to the first row:

5. Multiply the third line of the determinant by the reciprocal and add to the first line:

This determinant is the final result of the determinant, and its value is the value that is sought.

You're not talking about matrices (flanked by parentheses), but determinants (flanked by vertical bars) calculations. This method is called "determinant by one row (or column)". First of all, define it: cross out line i and j where aij is located

After the column, the product of the n-1 order determinant and the (i+j) power of -1 of the remaining elements in the original order is the algebraic remainder of aij. Then the value of the original determinant d=ai1*ai1+ai2*ai2+....+ain*ain.(i=1,2,…n) or d=a1j*a1j+a2j*a2j+....+anj*anj.

j+1,2，…，n)

In this problem, the original 5th-order determinant is pressed on the first line, because the first 4 of the first row are 0, so the first four terms are 0, so the determinant = 1*(-1) (1+5)*000

01 and then put this 4th order line 00

1-11-a111-a

40 columns according to the first line, arc Gong Gang clamp locust brother viagra company to get that negative 3-order determinant; Then press the 3rd order determinant on the first line, and then multiply it with the -1 in front of it to get the negative 2nd order determinant; The last 2nd order determinant = a11*a22-a12*a21=a-1, multiply the original negative sign in front to get the result (1-a).

In fact, this method is to gradually turn the high-order determinant into a simple low-order determinant to facilitate calculation, and the principle is very simple, and you can use it proficiently with more practice. Hope it helps

Subtract row 2 from row 3 and extract the common factor -8 from row 3 and add column 3 to column 2.

Then press line 3.

The 2nd order determinant is obtained, and then, factored out, it can be obtained.

ax = 2e, x = 2a^(-1)

a, e) =

The elementary row is transformed to.

The elementary row is transformed to.

The elementary row is transformed to.

a^(-1) =

x = 2a^(-1) =