In this topic, we will consider the concept of a matrix, as well as the types of matrices. Since there are a lot of terms in this topic, I will add summary to make it easier to navigate the material.
Definition of a matrix and its element. Notation.
Matrix is a table with $m$ rows and $n$ columns. Elements of a matrix can be objects of a completely diverse nature: numbers, variables, or, for example, other matrices. For example, the matrix $\left(\begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right)$ has 3 rows and 2 columns; its elements are integers. The matrix $\left(\begin(array) (cccc) a & a^9+2 & 9 & \sin x \\ -9 & 3t^2-4 & u-t & 8\end(array) \right)$ contains 2 rows and 4 columns.
Different ways to write matrices: show\hide
The matrix can be written not only in round brackets, but also in square or double straight brackets. Below is the same matrix in different notation:
$$ \left(\begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right);\;\; \left[ \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right]; \;\; \left \Vert \begin(array) (cc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right \Vert $$
The product $m\times n$ is called matrix size. For example, if the matrix contains 5 rows and 3 columns, then one speaks of a $5\times 3$ matrix. The matrix $\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ has size $3 \times 2$.
Matrices are usually denoted by capital letters of the Latin alphabet: $A$, $B$, $C$, and so on. For example, $B=\left(\begin(array) (ccc) 5 & 3 \\ 0 & -87 \\ 8 & 0 \end(array) \right)$. Line numbering goes from top to bottom; columns - from left to right. For example, the first row of the matrix $B$ contains elements 5 and 3, and the second column contains elements 3, -87, 0.
Elements of matrices are usually denoted by small letters. For example, the elements of the matrix $A$ are denoted by $a_(ij)$. The double index $ij$ contains information about the position of the element in the matrix. The number $i$ is the number of the row, and the number $j$ is the number of the column, at the intersection of which the element $a_(ij)$ is located. For example, at the intersection of the second row and fifth column of the matrix $A=\left(\begin(array) (cccccc) 51 & 37 & -9 & 0 & 9 & 97 \\ 1 & 2 & 3 & 41 & 59 & 6 \ \ -17 & -15 & -13 & -11 & -8 & -5 \\ 52 & 31 & -4 & -1 & 17 & 90 \end(array) \right)$ element $a_(25)= $59:
Similarly, at the intersection of the first row and the first column, we have the element $a_(11)=51$; at the intersection of the third row and the second column - the element $a_(32)=-15$ and so on. Note that $a_(32)$ is read as "a three two" but not "a thirty two".
For the abbreviated designation of the matrix $A$, the size of which is equal to $m\times n$, the notation $A_(m\times n)$ is used. The following notation is often used:
$$ A_(m\times(n))=(a_(ij)) $$
Here $(a_(ij))$ indicates the designation of the elements of the matrix $A$, i.e. says that the elements of the matrix $A$ are denoted as $a_(ij)$. In expanded form, the matrix $A_(m\times n)=(a_(ij))$ can be written as follows:
$$ A_(m\times n)=\left(\begin(array)(cccc) a_(11) & a_(12) & \ldots & a_(1n) \\ a_(21) & a_(22) & \ldots & a_(2n) \\ \ldots & \ldots & \ldots & \ldots \\ a_(m1) & a_(m2) & \ldots & a_(mn) \end(array) \right) $$
Let's introduce another term - equal matrices.
Two matrices of the same size $A_(m\times n)=(a_(ij))$ and $B_(m\times n)=(b_(ij))$ are called equal if their corresponding elements are equal, i.e. $a_(ij)=b_(ij)$ for all $i=\overline(1,m)$ and $j=\overline(1,n)$.
Explanation for the entry $i=\overline(1,m)$: show\hide
The entry "$i=\overline(1,m)$" means that the parameter $i$ changes from 1 to m. For example, the entry $i=\overline(1,5)$ says that the $i$ parameter takes the values 1, 2, 3, 4, 5.
So, for the equality of matrices, two conditions are required: the coincidence of sizes and the equality of the corresponding elements. For example, the matrix $A=\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$ is not equal to the matrix $B=\left(\ begin(array)(cc) 8 & -9\\0 & -87 \end(array)\right)$ because matrix $A$ is $3\times 2$ and matrix $B$ is $2\times 2$. Also the matrix $A$ is not equal to the matrix $C=\left(\begin(array)(cc) 5 & 3\\98 & -87\\8 & 0\end(array)\right)$ because $a_( 21)\neq c_(21)$ (i.e. $0\neq 98$). But for the matrix $F=\left(\begin(array)(cc) 5 & 3\\0 & -87\\8 & 0\end(array)\right)$, we can safely write $A=F$ because both the sizes and the corresponding elements of the matrices $A$ and $F$ coincide.
Example #1
Determine the size of the matrix $A=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \\ -6 & 8 & 23 \\ 11 & -12 & -5 \ \ 4 & 0 & -10 \\ \end(array) \right)$. Specify what the elements $a_(12)$, $a_(33)$, $a_(43)$ are equal to.
This matrix contains 5 rows and 3 columns, so its size is $5\times 3$. The notation $A_(5\times 3)$ can also be used for this matrix.
The element $a_(12)$ is at the intersection of the first row and the second column, so $a_(12)=-2$. The element $a_(33)$ is at the intersection of the third row and the third column, so $a_(33)=23$. The element $a_(43)$ is at the intersection of the fourth row and the third column, so $a_(43)=-5$.
Answer: $a_(12)=-2$, $a_(33)=23$, $a_(43)=-5$.
Types of matrices depending on their size. Main and side diagonals. Matrix trace.
Let some matrix $A_(m\times n)$ be given. If $m=1$ (the matrix consists of one row), then the given matrix is called matrix-row. If $n=1$ (the matrix consists of one column), then such a matrix is called column matrix. For example, $\left(\begin(array) (ccccc) -1 & -2 & 0 & -9 & 8 \end(array) \right)$ is a row matrix, and $\left(\begin(array) (c) -1 \\ 5 \\ 6 \end(array) \right)$ - column matrix.
If the condition $m\neq n$ is true for the matrix $A_(m\times n)$ (that is, the number of rows is not equal to the number of columns), then it is often said that $A$ is a rectangular matrix. For example, the matrix $\left(\begin(array) (cccc) -1 & -2 & 0 & 9 \\ 5 & 9 & 5 & 1 \end(array) \right)$ has size $2\times 4$, those. contains 2 rows and 4 columns. Since the number of rows is not equal to the number of columns, this matrix is rectangular.
If the condition $m=n$ is true for the matrix $A_(m\times n)$ (ie, the number of rows is equal to the number of columns), then $A$ is said to be a square matrix of order $n$. For example, $\left(\begin(array) (cc) -1 & -2 \\ 5 & 9 \end(array) \right)$ is a second-order square matrix; $\left(\begin(array) (ccc) -1 & -2 & 9 \\ 5 & 9 & 8 \\ 1 & 0 & 4 \end(array) \right)$ is a 3rd order square matrix. IN general view the square matrix $A_(n\times n)$ can be written as follows:
$$ A_(n\times n)=\left(\begin(array)(cccc) a_(11) & a_(12) & \ldots & a_(1n) \\ a_(21) & a_(22) & \ldots & a_(2n) \\ \ldots & \ldots & \ldots & \ldots \\ a_(n1) & a_(n2) & \ldots & a_(nn) \end(array) \right) $$
The elements $a_(11)$, $a_(22)$, $\ldots$, $a_(nn)$ are said to be on main diagonal matrices $A_(n\times n)$. These elements are called main diagonal elements(or just diagonal elements). The elements $a_(1n)$, $a_(2 \; n-1)$, $\ldots$, $a_(n1)$ are on side (secondary) diagonal; they are called secondary diagonal elements. For example, for the matrix $C=\left(\begin(array)(cccc)2&-2&9&1\\5&9&8& 0\\1& 0 & 4 & -7 \\ -4 & -9 & 5 & 6\end(array) \right)$ we have:
The elements $c_(11)=2$, $c_(22)=9$, $c_(33)=4$, $c_(44)=6$ are the main diagonal elements; the elements $c_(14)=1$, $c_(23)=8$, $c_(32)=0$, $c_(41)=-4$ are secondary diagonal elements.
The sum of the main diagonal elements is called followed by a matrix and denoted by $\Tr A$ (or $\Sp A$):
$$ \Tr A=a_(11)+a_(22)+\ldots+a_(nn) $$
For example, for the matrix $C=\left(\begin(array) (cccc) 2 & -2 & 9 & 1\\5 & 9 & 8 & 0\\1 & 0 & 4 & -7\\-4 & -9 & 5 & 6 \end(array)\right)$ we have:
$$ \Tr C=2+9+4+6=21. $$
The concept of diagonal elements is also used for non-square matrices. For example, for the matrix $B=\left(\begin(array) (ccccc) 2 & -2 & 9 & 1 & 7 \\ 5 & -9 & 8 & 0 & -6 \\ 1 & 0 & 4 & - 7 & -6 \end(array) \right)$ the main diagonal elements will be $b_(11)=2$, $b_(22)=-9$, $b_(33)=4$.
Types of matrices depending on the values of their elements.
If all elements of the matrix $A_(m\times n)$ are equal to zero, then such a matrix is called null and is usually denoted by the letter $O$. For example, $\left(\begin(array) (cc) 0 & 0 \\ 0 & 0 \\ 0 & 0 \end(array) \right)$, $\left(\begin(array) (ccc) 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end(array) \right)$ are zero matrices.
Consider some non-zero row of the matrix $A$, i.e. a string that contains at least one non-zero element. leading element of a non-zero string, let's call it the first (counting from left to right) non-zero element. For example, consider the following matrix:
$$W=\left(\begin(array)(cccc) 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 12\\ 0 & -9 & 5 & 9 \end(array)\right)$ $
In the second line, the fourth element will be leading, i.e. $w_(24)=12$, and in the third line the leading element will be the second element, i.e. $w_(32)=-9$.
The matrix $A_(m\times n)=\left(a_(ij)\right)$ is called stepped if it satisfies two conditions:
- Null rows, if any, are located below all non-null rows.
- The numbers of leading elements of non-zero strings form a strictly increasing sequence, i.e. if $a_(1k_1)$, $a_(2k_2)$, ..., $a_(rk_r)$ are leading elements of nonzero rows of matrix $A$, then $k_1\lt(k_2)\lt\ldots\lt( k_r)$.
Examples of step matrices:
$$ \left(\begin(array)(cccccc) 0 & 0 & 2 & 0 & -4 & 1\\ 0 & 0 & 0 & 0 & -9 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end(array)\right);\; \left(\begin(array)(cccc) 5 & -2 & 2 & -8\\ 0 & 4 & 0 & 0\\ 0 & 0 & 0 & -10 \end(array)\right). $$
For comparison: matrix $Q=\left(\begin(array)(ccccc) 2 & -2 & 0 & 1 & 9\\0 & 0 & 0 & 7 & 9\\0 & -5 & 0 & 10 & 6\end(array)\right)$ is not a step matrix, since the second condition in the definition of a step matrix is violated. The leading elements in the second and third lines $q_(24)=7$ and $q_(32)=10$ are numbered $k_2=4$ and $k_3=2$. For a step matrix, the condition $k_2\lt(k_3)$ must be satisfied, which is violated in this case. I note that if we swap the second and third rows, we get a stepped matrix: $\left(\begin(array)(ccccc) 2 & -2 & 0 & 1 & 9\\0 & -5 & 0 & 10 & 6 \\0 & 0 & 0 & 7 & 9\end(array)\right)$.
The step matrix is called trapezoidal or trapezoidal, if the leading elements $a_(1k_1)$, $a_(2k_2)$, ..., $a_(rk_r)$ satisfy the conditions $k_1=1$, $k_2=2$,..., $k_r= r$, i.e. diagonal elements are leading. In general, a trapezoidal matrix can be written as follows:
$$ A_(m\times(n)) =\left(\begin(array) (cccccc) a_(11) & a_(12) & \ldots & a_(1r) & \ldots & a_(1n)\\ 0 & a_(22) & \ldots & a_(2r) & \ldots & a_(2n)\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & a_(rr) & \ldots & a_(rn)\\ 0 & 0 & \ldots & 0 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & 0 & \ldots & 0 \end(array)\right) $$
Examples of trapezoidal matrices:
$$ \left(\begin(array)(cccccc) 4 & 0 & 2 & 0 & -4 & 1\\ 0 & -2 & 0 & 0 & -9 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end(array)\right);\; \left(\begin(array)(cccc) 5 & -2 & 2 & -8\\ 0 & 4 & 0 & 0\\ 0 & 0 & -3 & -10 \end(array)\right). $$
Let us give some more definitions for square matrices. If all elements of a square matrix located under the main diagonal are equal to zero, then such a matrix is called upper triangular matrix. For example, $\left(\begin(array) (cccc) 2 & -2 & 9 & 1 \\ 0 & 9 & 8 & 0 \\ 0 & 0 & 4 & -7 \\ 0 & 0 & 0 & 6 \end(array) \right)$ - upper triangular matrix. Note that the definition of the upper triangular matrix does not say anything about the values of the elements located above the main diagonal or on the main diagonal. They may or may not be zero, it doesn't matter. For example, $\left(\begin(array) (ccc) 0 & 0 & 9 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end(array) \right)$ is also an upper triangular matrix.
If all elements of a square matrix located above the main diagonal are equal to zero, then such a matrix is called lower triangular matrix. For example, $\left(\begin(array) (cccc) 3 & 0 & 0 & 0 \\ -5 & 1 & 0 & 0 \\ 8 & 2 & 1 & 0 \\ 5 & 4 & 0 & 6 \ end(array) \right)$ - lower triangular matrix. Note that the definition of a lower triangular matrix does not say anything about the values of the elements below or on the main diagonal. They may or may not be null, it doesn't matter. For example, $\left(\begin(array) (ccc) -5 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 9 \end(array) \right)$ and $\left(\begin (array) (ccc) 0 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end(array) \right)$ are also lower triangular matrices.
The square matrix is called diagonal if all elements of this matrix that are not on the main diagonal are equal to zero. Example: $\left(\begin(array) (cccc) 3 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 \ end(array)\right)$. The elements on the main diagonal can be anything (equal to zero or not) - this is not essential.
The diagonal matrix is called single if all elements of this matrix located on the main diagonal are equal to 1. For example, $\left(\begin(array) (cccc) 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end(array)\right)$ - 4th order identity matrix; $\left(\begin(array) (cc) 1 & 0 \\ 0 & 1 \end(array)\right)$ is the second order identity matrix.
Note that the elements of the matrix can be not only numbers. Imagine that you are describing the books that are on your bookshelf. Let your shelf be in order and all the books stand in strictly defined places. The table that will contain the description of your library (according to the shelves and the sequence of books on the shelf) will also be a matrix. But such a matrix will not be numeric. Another example. Instead of numbers, there are different functions, united among themselves by some dependence. The resulting table will also be called a matrix. In other words, Matrix is any rectangular table made up of homogeneous elements. Here and below we will talk about matrices composed of numbers.
Instead of parentheses, matrices are written using square brackets or straight double vertical lines.
 |
(2.1*) |
Definition 2. If in the expression(1) m = n , then they talk about square matrix, and if , something about rectangular.
Depending on the values of m and n, there are some special types of matrices:
The most important characteristic square matrix is its determinant or determinant, which is composed of matrix elements and is denoted
Obviously, D E =1 ; .
Definition 3. If , then the matrix A called non-degenerate or not special.
Definition 4. If detA = 0 , then the matrix A called degenerate or special.
Definition 5. Two matrices A And B called equal and write A=B if they have the same dimensions and their corresponding elements are equal, i.e..
For example, the matrices and are equal, because they are equal in size and each element of one matrix is equal to the corresponding element of the other matrix. But the matrices cannot be called equal, although the determinants of both matrices are equal, and the dimensions of the matrices are the same, but not all elements in the same places are equal. The matrices are different because they have different sizes. The first matrix is 2x3 and the second 3x2. Although the number of elements is the same - 6 and the elements themselves are the same 1, 2, 3, 4, 5, 6, but they are in different places in each matrix. But the matrices and are equal, according to Definition 5.
Definition 6. If we fix a certain number of matrix columns A and the same number of its rows, then the elements at the intersection of the specified columns and rows form a square matrix n- th order, the determinant of which called minor k- th order matrix A.
Example. Write out three minors of the second order of the matrix
Ticket 17:
Question 1: Definition of a parabola. Derivation of the equation:
Definition. A parabola is a set of points in a plane, each of which is at the same distance from a given point, called the focus, and from a given straight line, called the directrix and not passing through the focus.
Let us place the origin of coordinates in the middle between the focus and the directrix.
The value p (the distance from the focus to the directrix) is called the parameter of the parabola. We derive the canonical equation of the parabola.
From geometric relationships: AM = MF; AM = x + p/2;
MF2 = y2 + (x – p/2)2
(x + p/2)2 = y2 + (x – p/2)2
x2 + xp + p2/4 = y2 + x2 – xp + p2/4
Directrix equation: x = -p/2.
Question 2: Cauchy's theorem:
Theorem: Let the functions and be differentiable on the interval and continuous for and , moreover, for all . Then there is a point in the interval such that
geometric sense : These theorems consist in the fact that inside there is a point t 0 , the slopes of which are calculated by equality:
Proof.
Let us first prove that 
, that is, that the fraction on the left side of the formula makes sense. Indeed, for this difference, we can write the formula for finite increments:
at some . But on the right side of this formula, both factors are nonzero.
To prove the theorem, we introduce an auxiliary function
The function is obviously differentiable for all and continuous at the points and , since the functions and have these properties. Moreover, it is obvious that for , we obtain . Let's show that and:
Hence, the function satisfies the conditions of Rolle's theorem on the interval. Therefore, there is a point such that .
We now calculate the derivative of the function:
We get that
whence we obtain the assertion of the theorem:
Comment: We can consider the functions and coordinates of a point moving on a plane, which describes a line connecting the starting point with the end point. (Then the equations and parametrically set some dependence, the graph of which is the line.)
Fig.5.6. The chord is parallel to some tangent to the curve
The ratio, as it is easy to see from the drawing, then sets the slope of the chord connecting the points and . At the same time, according to the formula for the derivative of a function given parametrically, we have: 
. So the fraction is the slope of the tangent to the line at some point 
. Thus, the statement of the theorem means, from a geometric point of view, that there is a point on the line such that the tangent drawn at this point is parallel to the chord connecting the extreme points of the line. But this is the same statement that constituted the geometric meaning of Lagrange's theorem. Only in Lagrange's theorem the line was given by an explicit dependence, and in Cauchy's theorem by a dependence given in parametric form.
Ticket 18:
Question 1: The concept of a matrix. Matrix classification:
Definition.
A matrix of size mn, where m is the number of rows, n is the number of columns, is a table of numbers arranged in a certain order. These numbers are called matrix elements. The place of each element is uniquely determined by the number of the row and column at the intersection of which it is located. Matrix elements are denoted by aij, where i is the row number and j is the column number. A = 
Matrix classification:.
A matrix can have one row or one column. Generally speaking, a matrix can even consist of one element.
Definition . If the number of columns of the matrix is equal to the number of rows (m=n), then the matrix is called square.
Definition
. View Matrix: 
= E is called the identity matrix.
Definition. If amn = anm , then the matrix is called symmetric. Example. - symmetric matrix
Definition
. Square view matrix 
called diagonal matrix
.
Question 2: Lagrange's theorem:
Theorem:
Let the function be differentiable on the interval and continuous at the points and . Then there is a point such that 
Geometric sense:
Let us first give a geometric illustration of the theorem. Connect the end points of the graph on the segment with a chord. Finite increments and 
are the lengths of the legs of the triangle, the hypotenuse of which is the drawn chord.
Fig.5.5. The tangent at some point is parallel to the chord
The ratio of the final increments and is the tangent of the slope of the chord. The theorem states that a tangent can be drawn to the graph of a differentiable function at some point, which will be parallel to the chord, that is, the slope of the tangent () will be equal to the angle chord slope (). But the presence of such a tangent is geometrically obvious.
Note that the drawn chord connecting the points and is the graph of a linear function. Since the slope of this linear function is obviously equal to 
, That 
Proof of Lagrange's theorem. We reduce the proof to the application of Rolle's theorem. To do this, we introduce an auxiliary function , that is,
notice, that 
and (by constructing the function ). Since a linear function is differentiable for all , the function thus satisfies all the properties listed in the condition of Rolle's theorem. Therefore, there is such a point that Cheat Sheet By Philosophy: Exam Ticket Answers Cheat sheet >> Philosophy
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A matrix is a special object in mathematics. It is depicted in the form of a rectangular or square table, composed of a certain number of rows and columns. In mathematics, there is a wide variety of types of matrices, differing in size or content. The numbers of its rows and columns are called orders. These objects are used in mathematics to organize the writing of systems linear equations and easy search of their results. Equations using a matrix are solved using the method of Carl Gauss, Gabriel Cramer, minors and algebraic additions, and many other ways. The basic skill when working with matrices is to reduce to However, first, let's figure out what types of matrices are distinguished by mathematicians.
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 from the element located in the upper left corner, and ends with the element in the lower right, respectively. 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 its 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 the square type. It is somewhat similar to the identity matrix, however, in its case, not all components of the main diagonal take on a 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 of a 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. A prerequisite is the following: if there is a zero row in the step matrix, then the remaining rows below it also do not contain numeric values.
Thus, we have considered the most important types of 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.
Only the last value remains - the 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 row.
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 casting it from the element located in the lower left corner - from the number 4. We need to turn this 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 not difficult 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 reduce a matrix to a stepped form, you first 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. In order to nullify the desired values of matrix tables, in some cases it is required to show imagination and correctly transform their columns or rows. Good luck in mathematics and in working with matrices!
Although researchers usually refer to classification as a means of predicting the class of "unknown" objects, we can also use it to test the accuracy of classification procedures. To do this, we take the "known" objects (which we used in the derivation of classifying functions) and apply the classification rules to them. The proportion of correctly classified objects indicates the accuracy of the procedure and indirectly confirms the degree of class separation. A table, or "classification matrix," can be drawn up describing the results. This will help us see which mistakes are being made more often.
Table 12. Classification matrix
Table 12 is a classification matrix for Senate voting data. Bardes' six variables correctly predict the factional distribution of all senators (except Capehart) whose factional affiliation is "known." The prediction accuracy in this case is 94.7% (the sum of correct predictions is 18 divided by the total number of "known" objects). We also see that the errors in this example are due to the poor separation of groups 1 and 4. In the bottom row of Table. 12 shows the distribution by groups of "unknown" objects. These are the senators whose factional affiliation Bardes was unable to determine from the data she had. Her main goal was to use discriminant analysis to classify the positions of these senators based on their votes, after which she continued to explore the attitude of the Senate towards various options for foreign aid.
The percentage of "known" objects that were classified correctly is an additional measure of differences between groups. We will use it, along with Wilks' general L-statistic and canonical correlations, to indicate the amount of discriminant information contained in the variables. As a direct measure of prediction accuracy, this percentage is the most appropriate measure of discriminant information. However, the value of the percentage can only be judged by the expected percentage of correct classifications when the distribution to classes was made randomly. If there are two classes, then with random classification, 50% correct predictions can be expected. For four classes, the expected accuracy is only 25%. If for two classes the classification procedure gives 60% correct predictions, then its efficiency is quite low, but for four classes the same result indicates significant efficiency, because a random classification would give only 25% correct predictions. This brings us to the -error statistic, which will be a standardized measure of performance for any number of classes:
where is the number of correctly classified objects, and is the a priori probability of belonging to the class.
The expression represents the number of objects that will be correctly predicted by randomly classifying them into classes in proportion to the a priori probabilities. If all classes are considered equal, then the prior probabilities are assumed to be equal to one divided by the number of classes. The maximum value of the -statistic is 1 and it is reached in the case of error-free prediction. A zero value indicates the inefficiency of the procedure, - statistics can also take negative values, which indicates poor discrimination or a degenerate case. Since it must be an integer, the numerator may become negative purely by chance, when there are no differences between classes.
