Triangular view of the matrix. Upper triangular matrix properties

Upper triangular matrix

triangular matrix is a square matrix in which all elements below or above the main diagonal are equal to zero.

An example of an upper triangular matrix

Upper triangular matrix is a square matrix in which all elements below the main diagonal are zero.

Lower triangular matrix is a square matrix in which all elements above the main diagonal are equal to zero.

Unitrian matrix(upper or lower) - a triangular matrix in which all elements on the main diagonal are equal to one.

Triangular matrices are used primarily in solving linear systems of equations when the matrix of the system is reduced to a triangular form using the following theorem:

Systems solution linear equations with a triangular matrix ( reverse stroke) is not difficult.

Properties

Triangular matrix determinant is equal to the product elements on its main diagonal.
The determinant of a unitriangular matrix is equal to one.
The set of nondegenerate upper triangular matrices of order n by multiplication with elements from the field k forms a group which is denoted UT(n, k) or UT n (k).
The set of nondegenerate lower triangular matrices of order n by multiplication with elements from the field k forms a group, which is denoted LT(n, k) or LT n (k).
The set of upper unitriangular matrices with elements from the field k forms a subgroup UT n (k) by multiplication, which is denoted SUT(n, k) or SUT n (k). An analogous subgroup of lower unitriangular matrices is denoted SLT(n, k) or SLT n (k).
The set of all upper triangular matrices with elements from the ring k forms an algebra with respect to the operations of addition, multiplication by ring elements, and matrix multiplication. A similar statement is true for lower triangular matrices.
Group UT n is solvable, and its unitriangular subgroup SUT n nilpotent.

Properties

The determinant of a triangular matrix is equal to the product of the elements on its main diagonal.
The determinant of a unitriangular matrix is equal to one.
The set of nondegenerate upper triangular matrices of order n by multiplication with elements from the field k forms a group which is denoted UT(n, k) or UT n (k).
The set of nondegenerate lower triangular matrices of order n by multiplication with elements from the field k forms a group, which is denoted LT(n, k) or LT n (k).
The set of upper unitriangular matrices with elements from the field k forms a subgroup UT n (k) by multiplication, which is denoted SUT(n, k) or SUT n (k). An analogous subgroup of lower unitriangular matrices is denoted SLT(n, k) or SLT n (k).
The set of all upper triangular matrices with elements from the ring k forms an algebra with respect to the operations of addition, multiplication by ring elements, and matrix multiplication. A similar statement is true for lower triangular matrices.
Group UT n is solvable, and its unitriangular subgroup SUT n nilpotent.

Zero type

All components of this kind of matrix are zeros. Meanwhile, the number of its rows and columns is absolutely different.

square type

The number of columns and rows of this type of matrix is the same. In other words, it is a "square" shape table. The number of its columns (or rows) is called the order. Special cases are the existence of a matrix of the second order (matrix 2x2), fourth order (4x4), tenth (10x10), seventeenth (17x17) and so on.

Column Vector

This is one of the simplest types of matrices, containing only one column, which includes three numerical values. It represents a number of free terms (numbers independent of variables) in systems of linear equations.

View similar to the previous one. Consists of three numerical elements, in turn organized in one line.

Diagonal type

Numerical values in the diagonal form of the matrix take only the components of the main diagonal (highlighted in green). The main diagonal starts with the element on the right upper corner, and ends with a number in the third column of the third row. The rest of the components are zero. The diagonal type is only a square matrix of some order. Among matrices of the diagonal form, one can single out a scalar one. All its components take the same values.

A subspecies of the diagonal matrix. All her numerical values are units. Using a single type of matrix tables, its basic transformations are performed or a matrix is found that is inverse to the original one.

Canonical type

The canonical form of the matrix is considered one of the main ones; casting to it is often needed to work. The number of rows and columns in the canonical matrix is different, it does not necessarily belong to square type. She is somewhat similar to identity matrix, however, in its case, not all components of the main diagonal take on the value equal to one. There can be two or four main diagonal units (it all depends on the length and width of the matrix). Or there may be no units at all (then it is considered zero). The remaining components of the canonical type, as well as the elements of the diagonal and unit types, are equal to zero.

triangular type

One of the most important types matrix, used when searching for its determinant and when performing simple operations. The triangular type comes from the diagonal type, so the matrix is also square. The triangular view of the matrix is divided into upper triangular and lower triangular.

In the upper triangular matrix (Fig. 1), only the elements that are above the main diagonal take on a value equal to zero. The components of the diagonal itself and the part of the matrix below it contain numerical values.

In the lower triangular matrix (Fig. 2), on the contrary, the elements located in the lower part of the matrix are equal to zero.

The form is necessary for finding the rank of a matrix, as well as for elementary operations on them (along with the triangular type). The step matrix is so named because it contains characteristic "steps" of zeros (as shown in the figure). In the stepped type, a diagonal of zeros is formed (not necessarily the main one), and all elements under this diagonal also have values equal to zero. The prerequisite is the following: if stepped matrix there is a null string, then the other strings below it also do not contain numeric values.

Thus, we have considered the most important types matrices needed to work with them. Now let's deal with the task of converting a matrix into the required form.

Reduction to a triangular form

How to bring the matrix to a triangular form? Most often, in assignments, you need to convert a matrix into a triangular form in order to find its determinant, otherwise called the determinant. When performing this procedure, it is extremely important to "preserve" the main diagonal of the matrix, because the determinant of a triangular matrix is exactly the product of the components of its main diagonal. Let me also remind you of alternative methods for finding the determinant. The square-type determinant is found using special formulas. For example, you can use the triangle method. For other matrices, the method of decomposition by row, column, or their elements is used. You can also apply the method of minors and algebraic complements of the matrix.

Let us analyze in detail the process of bringing a matrix to a triangular form using examples of some tasks.

Exercise 1

It is necessary to find the determinant of the presented matrix, using the method of bringing it to a triangular form.

The matrix given to us is a square matrix of the third order. Therefore, to transform it into a triangular form, we need to vanish two components of the first column and one component of the second.

To bring it to a triangular form, we start the transformation from the lower left corner of the matrix - from the number 6. To turn it to zero, we multiply the first row by three and subtract it from the last row.

Important! The top line does not change, but remains the same as in the original matrix. You do not need to write a string four times the original one. But the values of the rows whose components need to be set to zero are constantly changing.

Remains only last value- element of the third row of the second column. This is the number (-1). To turn it to zero, subtract the second from the first line.

Let's check:

detA = 2 x (-1) x 11 = -22.

Hence, the answer to the task: -22.

Task 2

It is necessary to find the determinant of the matrix by bringing it to a triangular form.

The presented matrix belongs to the square type and is a matrix of the fourth order. This means that it is necessary to vanish three components of the first column, two components of the second column, and one component of the third.

Let's start bringing it from the element located in the lower left corner - from the number 4. We need to reverse given number to zero. The easiest way to do this is to multiply the top row by four and then subtract it from the fourth row. Let us write down the result of the first stage of the transformation.

So, the component of the fourth row is set to zero. Let's move on to the first element of the third line, to the number 3. We perform a similar operation. Multiply by three the first row, subtract it from the third row and write the result.

We managed to set to zero all the components of the first column of this square matrix, with the exception of the number 1, an element of the main diagonal that does not require transformation. Now it is important to keep the resulting zeros, so we will perform transformations with rows, not columns. Let's move on to the second column of the presented matrix.

Let's start from the bottom again - from the element of the second column of the last row. This is the number (-7). However, in this case it is more convenient to start with the number (-1) - the element of the second column of the third row. To turn it to zero, subtract the second row from the third row. Then we multiply the second row by seven and subtract it from the fourth. We got zero instead of the element located in the fourth row of the second column. Now let's move on to the third column.

In this column, we need to turn to zero only one number - 4. This is easy to do: just add the third to the last row and see the zero we need.

After all the transformations, we brought the proposed matrix to a triangular form. Now, to find its determinant, you only need to multiply the resulting elements of the main diagonal. We get: detA = 1 x (-1) x (-4) x 40 = 160. Therefore, the solution is the number 160.

So, now the question of bringing the matrix to a triangular form will not make it difficult for you.

Reduction to stepped form

For elementary operations on matrices, the stepped form is less "demanded" than the triangular one. It is most commonly used to find the rank of a matrix (i.e., the number of its non-zero rows) or to determine linearly dependent and independent rows. However, the stepped view of the matrix is more versatile, as it is suitable not only for the square type, but for everyone else.

To bring the matrix to stepped view, first you need to find its determinant. For this, the above methods are suitable. The purpose of finding the determinant is to find out if it can be converted into a step matrix. If the determinant is greater or less than zero, then you can safely proceed to the task. If it is equal to zero, it will not work to reduce the matrix to a stepped form. In this case, you need to check if there are any errors in the record or in the matrix transformations. If there are no such inaccuracies, the task cannot be solved.

Let's consider how to bring the matrix to a stepped form using examples of several tasks.

Exercise 1. Find the rank of the given matrix table.

Before us is a square matrix of the third order (3x3). We know that in order to find the rank, it is necessary to reduce it to a stepped form. Therefore, we first need to find the determinant of the matrix. Let's use the triangle method: detA = (1 x 5 x 0) + (2 x 1 x 2) + (6 x 3 x 4) - (1 x 1 x 4) - (2 x 3 x 0) - (6 x 5 x 2) = 12.

Determinant = 12. It is greater than zero, which means that the matrix can be reduced to a stepped form. Let's start transforming it.

Let's start it with the element of the left column of the third row - the number 2. We multiply the top row by two and subtract it from the third one. Thanks to this operation, both the element we need and the number 4 - the element of the second column of the third row - turned into zero.

We see that as a result of the reduction, a triangular matrix was formed. In our case, the transformation cannot be continued, since the remaining components cannot be turned to zero.

So, we conclude that the number of rows containing numerical values in this matrix (or its rank) is 3. Answer to the task: 3.

Task 2. Determine the number of linearly independent rows of the given matrix.

We need to find such strings that cannot be converted to zero by any transformations. In fact, we need to find the number of non-zero rows, or the rank of the represented matrix. To do this, let's simplify it.

We see a matrix that does not belong to the square type. It has dimensions 3x4. Let's start the cast also from the element of the lower left corner - the number (-1).

Further transformations are not possible. So, we conclude that the number of linearly independent lines in it and the answer to the task is 3.

Now bringing the matrix to a stepped form is not an impossible task for you.

On the examples of these tasks, we analyzed the reduction of a matrix to a triangular form and a stepped form. To nullify desired values matrix tables, individual cases you need to show imagination and correctly transform their columns or rows. Good luck in mathematics and in working with matrices!

Page 2

A triangular matrix is a matrix in which all elements on one side of the main or secondary diagonal are equal to zero. What is the determinant of a triangular matrix.

Operations for performing the forward move of the Gauss method in accordance with the theorems of linear algebra do not change the value of the determinant. Obviously, the determinant of a triangular matrix is equal to the product of its diagonal elements.

This intuitive representation finds in some cases an exact quantitative expression. For example, we know (see (6) from § 1) that the determinant of a triangular matrix (upper or lower) is equal to the product of the elements on the main diagonal.

Triangular matrices have many remarkable properties, due to which they are widely used in the construction of most various methods solving algebra problems. So, for example, for square matrices the sum and product of triangular matrices of the same name is a triangular matrix of the same name, the determinant of a triangular matrix is equal to the product of the diagonal elements, eigenvalues of a triangular matrix coincide with its diagonal elements, a triangular matrix is easily inverted and its inverse will also be triangular.

It has already been noted earlier that the direct determination of the determinant requires a large amount of computation. At the same time, the determinant of a triangular matrix is easily calculated: it is equal to the product of its diagonal elements.

How more zeros among the elements of the matrix A and the better they are located, the easier it is to calculate the determinant det A. This intuitive representation finds in some cases an exact quantitative expression. For example, we know (see (6) from § 1) that the determinant of a triangular matrix (upper or lower) is equal to the product of the elements on the main diagonal.

For example, multiplying a determinant by a scalar is equivalent to multiplying the elements of any row or any column of a matrix by that scalar. From equation (40) and from the fact that the expansion is applicable to algebraic addition just as for the determinant, it follows that the determinant of a triangular matrix is equal to the product of its diagonal elements.

This possibility stems from three basic properties determinants. Adding a multiple of one string to another does not change the determinant. Swapping two strings changes the sign of the determinant. The determinant of a triangular matrix is simply the product of its diagonal elements. DECOMP uses the last component of the pivot vector to put the value 1 there, if any even number permutations, and the value is 1 if odd. To get the determinant, this value must be multiplied by the product of the diagonal elements of the output matrix.

Triangular view of the matrix. Upper triangular matrix properties

Properties

see also

See what the "Upper Triangular Matrix" is in other dictionaries:

Properties

see also

See what the "Triangular matrix" is in other dictionaries:

Zero type

square type

Column Vector

Diagonal type

Canonical type

triangular type

Reduction to a triangular form

Exercise 1

Task 2

Reduction to stepped form