An array is a collection of a finite number of numeric or symbolic elements that have a unique name, ordered in some way and having specific addresses. The MathCAD package uses arrays of two most common types: one-dimensional (vectors), two-dimensional (matrices).
All the built-in tools of the MathCAD package intended for working with matrices are collected on the vector and matrix toolbar. You can set a matrix either by inserting a template of the required size from the matrix panel and filling it with numbers, or by assigning its value to each matrix element (usually this is done in a loop). If an element of an array (matrix) has several indices (the ordinal number of the element), then they are separated by commas. For matrix elements, the first is the row number, the second is the column number. Array indexing by default starts at zero, but you can control this process. The number of the first element is stored in the predefined variable ORIGIN, the value of which can be changed. Having met the assignment ORIGIN: \u003d k, MathCAD will number all the arrays found below, starting from the number k. Any reference to an element with a lower number will generate an error message.
To enter a matrix (or vector), you need to do the following sequence of operations:
Set the name of the matrix and enter the assignment sign. For example, to set the matrix A, write A :. We get A: \u003d;
Using the Insert -\u003e Matrix ... command, or the Ctrl + M key combination, or by clicking on the Matrix panel button, filling in the array of empty fields for not too large arrays.
Also vectors and matrices can be specified using a discrete argument when there is some explicit dependence for calculating the elements through their indices.
Vector formation is carried out in the same way.
It should be noted the second option of forming matrices and vectors without accessing the window for working with matrices, but through variables with indices, for example, Ai, j, Bi. The index to the variable name is printed by pressing either the Xn button on the math toolbar or the [(open square bracket) key.
Operations with matrices and vectors are carried out according to the same rules as for arithmetic expressions.
ORIGIN: \u003d 1 define the number of the first element
form the matrix A
form the matrix B
we solve the matrix equation AX \u003d B
solution output
check
| A | \u003d -2 find the determinant
inverse matrix
ORIGIN: \u003d 0 define the number of the first element
А0,0: \u003d 1 А0,1 \u003d 1 form matrix А
A0.0: \u003d 5 A0.1 \u003d 3
B0: \u003d 138 B1 \u003d 540 form the matrix B
X: \u003d lsole (A, B) solve the matrix equation AX \u003d B
X0 \u003d 63 X1 \u003d 75 solution output
A0.0X0 + A0.1X1-B0 \u003d 0 check
A1.0X0 + A1.1X1-B1 \u003d 0.
Calculations with vectors and matrices.
Vector matrix operators.
To work with vectors and matrices, the MathCAD system contains a number of operators and functions. Let us introduce the following notation: for vectors - V, for matrices - M, and for scalars - Z.
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Operator |
Operator appointment |
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Addition of two vectors V1 and V2 |
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Subtraction of two vectors V1 and V2 |
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Change of sign of elements of vector V |
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Sign change for elements of matrix M |
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Subtracting the scalar Z from the vector V |
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Multiplication of a vector V by a scalar Z |
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Multiplication of two vectors V1 and V2 |
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Multiplication of matrix M by vector V |
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Multiplication of two matrices M1 and M2 |
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Division of vector V by scalar Z |
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Division of matrix M by scalar Z |
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Inversion of the matrix M |
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Raising a matrix M to the power n |
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Calculating the square root of mV |
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Calculation of the determinant of the matrix M |
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Transpose vector V |
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Transpose matrix M |
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Cross - multiplication of two vectors V1 and V2 |
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Obtaining a complex conjugate vector |
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Obtaining a complex conjugate matrix |
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Calculation of the sum of the elements of the vector V |
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Vectorization of vector V |
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Vectorization of matrix M |
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Selection of the nth column of the matrix M |
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Selection of the nth element of the vector V |
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Selection of the element (m, n) of the matrix M |
excel filter string matrix
The concept of "vectorization" means the simultaneous performance of mathematical operations in their scalar value on all elements of a vector or matrix, marked with vectorization. This can also be understood as the possibility of parallel computing.
If A and B are vectors, then A * B gives the dot product of these vectors. But the same product under the vectorization sign creates a new vector, each j-th element of which is the product of the j -th elements of vectors A and B. Vectorization allows using scalar operators and functions with arrays.
Vector and matrix functions.
There are also a number of built-in vector and matrix functions:
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returns the length of the vector |
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returns the index of the last element |
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returns the highest-valued element |
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returns the smallest element |
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returns a vector of real parts of a vector with complex elements |
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returns a vector of imaginary parts of a vector with complex elements |
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fully asymmetric tensor of dimension three. i, j, k must be integers between 0 and 2 (or between\u003e ORIGIN and ORIGIN + 2 if ORIGIN? 0). The result is 0 if any two arguments are equal, 1 if three arguments are an even permutation (0, 1, 2), and minus 1 if three arguments are a permutation (0, 1, 2) multiple of 2 and non-multiple of 4 |
There are also a number of built-in functions for working with matrices. They are listed below:
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Augment (M1, M2) |
Combines into one matrices М1 and М2, which have the same number of rows (the union is "side by side") |
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Creates the n * n identity square matrix |
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Combines into one matrices M1 and M2, which have the same number of columns, placing M1 over M2 |
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submatrix (A, ir, jr, ic, jc) |
Returns a submatrix consisting of all the elements in rows ir through jr and columns ic through jc (irJjr and icJjc) |
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Creates a diagonal matrix whose main diagonal element is the vector V |
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A matrix in which the (i, j) th element contains f (i, j), where i \u003d 0, 1,… m and j \u003d 0, 1,… n |
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Returns a matrix of real parts of a matrix M with complex elements |
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Returns a matrix of imaginary parts of a matrix M with complex elements |
Functions that return special characteristics of matrices.
The special characteristics of matrices are returned by the following functions:
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returns the number of columns of matrix M |
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returns the number of rows of the matrix M |
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returns the rank of the matrix M |
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returns the trace (sum of diagonal elements) of a square matrix M |
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returns the average of the elements of array M |
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returns the median of the elements of array M |
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returns the condition number of the matrix calculated in the L1 norm |
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returns the condition number of the matrix calculated in the L2 norm |
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Returns the condition number of a matrix, calculated in the Euclidean space norm |
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Returns the condition number of a matrix based on a uniform norm |
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Returns L1, the norm of the matrix M |
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Returns L2, the norm of the matrix M |
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Returns the Euclidean norm of the matrix M |
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Returns the undefined norm of the matrix M |
Additional matrix functions.
A number of additional matrix functions are included in professional versions of MathCAD:
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returns a vector containing the eigenvalues \u200b\u200bof the matrix M |
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for the specified matrix M and the specified eigenvalue Z, returns the vector belonging to this eigenvalue |
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returns a matrix whose columns are the eigenvectors of M (the order of the eigenvectors corresponds to the order of the eigenvalues \u200b\u200breturned by the eigenvals function) |
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returns a vector of generalized eigenvalues \u200b\u200bv corresponding to a solution to equation M? x \u003d vi - N - x (matrices M and N must be real) |
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returns a matrix whose columns contain normalized generalized eigenvectors |
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performs a triangular decomposition of the matrix M: P? M \u003d L? U, L and U are respectively lower and upper triangular matrices. All four matrices are square, of the same order |
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gives the decomposition of the matrix A, A \u003d Q? R, where Q is an orthogonal matrix and\u003e is an upper triangular matrix |
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gives a singular value decomposition of the matrix A of size n? m: A \u003d U? S VT where and are orthogonal matrices of size m m and n n, respectively, S is a diagonal matrix on the diagonal of which the singular values \u200b\u200bof the matrix A are located |
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returns a vector containing singular values \u200b\u200bof matrix A of size m n, where mі n |
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returns the matrix left inverse to matrix A. L A \u003d E, where E is the nn identity matrix, L is an n m rectangular matrix, A is an m n rectangular matrix |
Sorting functions for vectors and matrices.
Starting with the third version, some additional sorting functions appeared in the MathCAD system - permutations of elements of vectors and matrices.
Consider the simplest matrix algebra operations implemented in MathCAD in the form of operators, and it should be noted that their recording is as close as possible to the mathematical form of recording. The most frequently used operations are located on the toolbar Matrix (Matrix) (fig. 14), the rest can be found using the menu Insert → Function... function categories Vector and Matrix.
Figure: 14. Toolbars Matrix and Logic
Transpose is called an operation that translates a matrix of dimension M× N into the dimension matrix N× M, making the columns of the original matrix rows and the rows columns. Entering the transpose character is done using the toolbar Matrix(Matrix) or keystrokes
Addition and subtraction... IN MathCAD you can both add matrices and subtract them from each other. For these operators, the standard characters "+" or "-" are used, respectively. Matrices must have the same dimension, otherwise an error message will be displayed. Each element of the sum of two matrices is equal to the sum of the corresponding elements of the matrix terms. The result of a unary operation of changing the sign of a matrix is \u200b\u200bequivalent to changing the sign of all its elements. In order to change the sign of a matrix, it is enough to enter a minus sign in front of it, as in front of an ordinary number.
When multiplication it should be remembered that the dimension matrix M× N it is permissible to multiply only by a matrix of dimension N× P (P can be anything). The result is a matrix of dimension M× P.
To enter the multiplication symbol, you need to press the key with an asterisk<*> or use the toolbar Matrix(Matrix) by clicking the button on it Dot Product (Multiplication). Matrix multiplication is indicated by a dot by default.
The following built-in functions are provided to obtain information about the characteristics of matrices or vectors:
· rows (A) - number of lines;
· cols (A) - the number of columns;
· length (v) - the number of vector elements;
· last (v) - index of the last element of the vector,
where A is a matrix or vector; v is a vector.
Scalar product vectors (vector inner product)
is defined as a scalar equal to the sum of the pairwise products of the corresponding elements. Vectors must have the same dimension, the dot product has the same dimension. Dot product of two vectors u and v equally 
, where is the angle between vectors. If the vectors are orthogonal, then their dot product is zero. The dot product is denoted by the same symbol as multiplication.
Vector product (cross product) of two vectors u and v with the angle between them is equal to the vector with the modulus 
directed perpendicular to the plane of vectors u and v... The cross product is denoted by the symbol
Determinant matrices are denoted by a standard mathematical symbol. To enter an operator for finding the determinant of a matrix, you can click Determinant (Qualifier) \u200b\u200bon the toolbar Matrix(Matrix) or type on the keyboard<|
\u003e (pressing the keys
By rank (rank) matrices are called the largest natural number kfor which there is a nonzero determinant k-th order submatrix composed of any intersection k columns and k rows of the matrix. To determine the rank of a matrix in MathCAD function used rank (A), where A is the matrix whose rank is to be found.
As you know, the search inverse matrix is possible if the matrix is \u200b\u200bsquare and its determinant is not zero. The product of the original matrix by its inverse is by definition the identity matrix. To enter the search operator for the inverse matrix, press the button Inversion (Inverse) on the toolbar Matrix(Matrix).
Linear algebra uses various vector and matrix norms (norm), which associate a matrix with some scalar numerical characteristic. The matrix norm reflects the order of magnitude of the matrix elements. In different specific problems of linear algebra, different types of norms are applied. MathCAD has four built-in functions for calculating different norms for square matrices:
· norm1 (A) - norm in the space L1;
· norm2 (A) - norm in the space L2;
· norme (A) - the euclidean norm;
· normi (A) - max-norm, or -norm (infinity norm):
where A is a square matrix.
It is often necessary to rearrange the elements of a matrix or vector by placing them in a particular row or column in ascending or descending order. For this, there are several built-in functions that allow flexible control over the sorting of matrices:
· sort (v) - sorting of vector elements in ascending order;
· reverse (v) - permutation of vector elements in reverse order;
· csort (A, i) - sorting the rows of the matrix by arranging the elements of the i-column in ascending order;
· rsort (A, i) - sorting the columns of the matrix by arranging the element of the i-th row in ascending order, where v - vector; A - matrix; i is the index of the row or column.
Examples of the operation of the above operators are shown in Fig. 15.
To set logical functions in MathCAD there is a toolbar brain teaser (Boolean) fig. 6.13. It contains buttons that reflect relationships (\u003d,\u003e,<, £, ³, ¹) и основные логические операции: conjunction , disjunction , negation and exclusive or (XOR). As you know, all logical functions can be expressed through three main ones: conjunction, disjunction and negation, which is reflected in the listing of the MathCAD program in Fig. 16. Also in MathCAD you can interpret and complex logical functions (Fig. 16).
Figure: 15. Working with matrices in MathCAD
Figure: 16. Logical functions in MathCAD
Hello again. You and I managed to finish quite successfully with school algebra, and, therefore, we can move on to more complex things - for example, to that branch of mathematics that is usually called linear algebra in universities. Of course, this section is very, very extensive, and there is no need to even think about how to cover at least most of it. Nevertheless, you and I, I think, are almost entirely capable of learning to use some of it - for example, we can learn to work with vectors and matrices in MathCAD. These two concepts are the most important mathematical abstractions, the role of which is difficult to overestimate in scientific and technological progress. After all, it was the use of vectors and vector analysis that at one time allowed Oliver Heaviside to reduce the number of Maxwell's equations describing the electromagnetic field, from two dozen to only four. Before vector analysis, I think we’ll get there in due time, but for now we’ll deal with more prosaic things. Which ones exactly? I think everything will become clear if you continue reading this article.
A little about matrix elements
Although in what follows we will deal with vectors and matrices, for the sake of brevity I will just say “matrix”, implying that by a vector we mean a special case of a matrix, namely, its kind, which is a single column. If the vector is a string, then this will be specially stipulated. However, I think it will hardly come to this. Generally speaking, we are already a little able to work with matrices. Well, it’s not that it’s right to work like this - at least, we have already introduced them into MathCAD. Nevertheless, I think it will not be superfluous to remind you that you can enter a matrix into the workspace using the Matrix or Vector button located on the Matrix panel, or using the Ctrl + M hotkey combination. However, we have not yet touched on one small, but extremely important point, namely, how to address in calculations not to the whole matrix, but to its individual elements. This is actually very simple to do. Suppose we are given a square matrix X of two by two elements. The top-left element will be at indices 0,0; the lower right will have indices of 1.1, respectively. That is, as you can see, the elements of the matrix are numbered from zero. This, in general, is quite convenient, however, if you are more accustomed to numbering them from one, or, say, from 1024, then you can change the value of the built-in variable ORIGIN by entering MatchCAD in a string at the very beginning of the document (without quotes) ... Or, respectively, "ORIGIN: \u003d 1024". You can change the value of the variable in the MathCAD options window by selecting the Worksheet options item from the Tools menu and changing the value of the ORIGIN variable to the one you need on the Built-In Variables tab.
So, back to our matrix elements. To "take out" the first element from it, you need to write the following: X0, 0. In order to write the indices at the bottom of the name of the variable that denotes the matrix, you can use the Subscript button from the same toolbar for matrix and vector calculations or from the keyboard go to the subscript using the "[" (Russian "x") key. Note that a comma is used to separate row and column indices. Matrix elements can not only be extracted from a matrix specified in a table. You can specify several elements with the corresponding indices during the calculation, and after that MathCAD will independently form a matrix from them (but only when you have already specified all of its elements - otherwise, the undefined elements will be assigned zero values). You can use ranked variables to define the elements of a matrix. At times it is not only convenient, but very convenient. So, for example, you can set the following formula for the matrix X, which describes the values \u200b\u200bof each of its elements in the i-th row and j-th column: Xi, j: \u003d i * j. Before such a definition of the matrix elements, it remains only to determine the range in which i and j will change. For example, I took the values \u200b\u200bi: \u003d 0..5 and j: \u003d 0..5, but, of course, you can set any other range you need depending on the requirements of your problem solved using matrices in the MathCAD environment.
Matrix operations
Of course, there are many interesting and not very interesting aspects of using matrices in real problems, but sooner or later they all run into the need to carry out simple algebraic operations with matrices. Carrying out them manually is a laborious task, and you can spend time with much greater benefit by shifting the routine work to MathCAD. To begin with, let's get acquainted with those functions that actually do not perform any mathematical operations, but at the same time are very important in operations with matrices. These functions allow you to combine two matrices into one (do not add matrices, but simply combine their elements) and select another matrix from the matrix. The first of the functions is augment. It combines two matrices with the same number of rows, in such a way that one of them is formed, in which the elements of these two are located, as they say, "shoulder to shoulder." This function only needs two matrices to be merged as arguments. Its counterpart for those matrices that have the same number of columns and must be combined one above the other is the stack function. Its arguments must also be two merged matrices. The function, which does not unite, but, on the contrary, “cuts” matrices, is called submatrix. For it, you need to specify the name of the matrix from which we want to select the submatrix, and the coordinates of the elements of the new matrix in the old matrix. That is, in order to cut a 4x4 matrix from the upper left elements of a 5x5 matrix, we need to call this function with the following parameters: submatrix (Y, 0, 4, 0, 4). Here Y is, of course, the name of a 5x5 matrix. A demonstration of the use of all these functions in relation to specific matrices can be seen in the corresponding illustration to the article.
Now, I think so, it's time to move on to arithmetic operations on matrices. You will see that using them in MathCAD does not require any special knowledge of linear algebra from you - apart from, of course, a general idea of \u200b\u200bhow matrix operations work in order to understand what they might end up with. MathCAD is good for the user in that it allows him to work with vectors and matrices in the same way as with ordinary scalars, that is, variables containing only and exclusively real numbers. Try to define two matrices (I called them aa and bb), and then apply the addition operation to them in the same way as you once applied it to ordinary numbers. Of course, in order for the matrices to be stacked, they must have the same dimensions. Similarly, you can try subtracting another from one matrix or multiplying them. You will see that MathCAD successfully copes with such tasks without straining the user with unnecessary calculations.
Transpose matrices in MathCAD is no more difficult than adding or multiplying them. Calculate inverse matrices, however, too. In all these tasks, the corresponding operators with will help. They are indicated on it with the same icons as in textbooks on linear algebra, and therefore for a user who is at least minimally familiar with matrix calculus, everything should be simple and clear. You just need to remember that you cannot calculate the inverse matrix for a matrix that is degenerate (that is, it has zero or proportional rows or columns to each other). In order not to dwell unnecessarily on these simple operations, I will simply give an illustration demonstrating their practical use. I think the readers will not have any problems with either transposition or calculating the inverse matrix using MathCAD.
Another frequently performed operation is the calculation of the determinant, or determinant of a matrix. I think that I will not upset you at all with the news that in MathCAD and with determinants it is just as easy to work as with everything else related to matrices. The Determinant button is responsible for its calculation, located, of course, on the Matrix panel. Since the determinant in MathCAD, as well as in linear algebra in general, is denoted using the symbols of straight lines that limit the matrix (or the name of the variable denoting it), it is quite logical that you can insert these very straight lines into the text of the expression using the corresponding key on the keyboard: Shift + \\. I think that this simple combination will still be more convenient to use than to search for the necessary button on the MathCAD toolbar every time.
Matrix appearance
Of course, we are far from finishing our acquaintance with matrices in MathCAD, however, I think, for today, direct mathematics is enough. At the end of the article, I'll better talk about how you can change the way matrices are displayed in the MathCAD environment - perhaps some of the readers of this article will find this information useful and interesting.
The fact is that MathCAD can display matrices not only in the usual form of numbers enclosed in brackets, but also in the form of tables. If you use this mathematical package for any statistical calculations and at the same time work with large data sets, then, of course, such a matrix display will be more preferable for you than the traditional one. To change the way matrices are displayed, double-click on the required matrix and in the window that appears, go to the Display Options tab. Next, in the Matrix display style field, select the Table value. The matrix will look exactly the same as in the corresponding illustration for this article.
The appearance of the table can also be further customized by right-clicking on it and selecting Properties. In the window that appears, you can remove the checkbox from the Show column / row labels item to hide the display of row and column numbering in the matrix, represented as a table. On the Data Range tab, you can select the range of displayed rows and columns of the matrix, which is also useful for matrices that contain a large number of elements.
Well, that's enough for today, I think. This is just the beginning of working with matrices - next time we'll talk about much more interesting things than just addition, transposition and calculating the determinant.
Consider a few more extremely important actions of linear algebra related to the concept of a matrix determinant. Despite the fact that some of them are also implemented in Mathcad in the form of operators, they require (when performing calculations using numerical algorithms) incomparably more attention than the operators mentioned in the two previous sections.
Determinant ( Determinant) of a matrix is \u200b\u200bdenoted by a standard mathematical symbol. To enter an operator for finding the determinant of a matrix, you can click Determinant (Qualifier) \u200b\u200bon the toolbar Matrix (Matrix) (Listing 7.14) or type on the keyboard | (pressing the keys SHIFT + \\). Any of these actions results in a placeholder in which to place the matrix. To calculate the determinant of an already entered matrix:
- Move the cursor in the document in such a way to place the matrix between the input lines (recall that the input lines are vertical and horizontal lines of blue, forming a corner indicating the current editing area).
- Enter the operator to find the determinant of the matrix.
- Enter an equal sign (or symbolic output) to compute the determinant (numerically or analytically, respectively, as shown in Listing 7.14).
Attention!
Do not confuse the operators for calculating the determinant of a square matrix and the length of a vector. Matncad 12 introduces forced control of user actions when entering these operators to avoid confusion (since the same character is used for these two operations). When trying to compute the determinant of a matrix using the operator | A |entered from the panel Calculator (Calculator), not Matrix (Matrix), an error message will be displayed, and the result of calculating the determinant will appear only after the user invokes the context menu and confirms in it that he is going to calculate the determinant of the matrix. The same goes for the length of the vector, if you try to enter it not from the panel Calculator (Calculator), and from the panel Matrix (Matrix).
You may have come across these concepts while working in Excel - a column of numbers is called a column vector, a row is a row vector. The block of objects is a matrix. Calculations in Excel are essentially vector and matrix operations. In this lesson, we will get acquainted with similar calculations in Mathcad, and we will understand why they are easier to do in Mathcad.
Introduction
In the previous tutorials, our vectors started at element number "0". In this tutorial, let's make the first element number "1" for simplicity. This can be done using the Calculation tab -\u003e Document parameters -\u003e ORIGIN:
This value can be printed directly into the document so as not to forget it and not get confused:
Now let's look at a few matrices:
As you can see, they can include numbers, symbols, and even functions. They can also contain text elements (strings).
A matrix element can be displayed using subscripts:
The matrices above are 2x2 square, but they can be of any size in rows and columns:
Remember: the first number is the row number (or their number), the second is the column.
Subscripted items:
For a column vector, the second index can be omitted, but not for a row vector:
In the Mathematics tab -\u003e Operators and Symbols -\u003e Operators -\u003e Vectors and Matrices, you will find commands for highlighting columns and rows:
Many operations for vectors and matrices are similar to operations for ordinary numbers, variables and functions: addition, subtraction, some types of multiplication. Finding an inverse matrix is \u200b\u200bclose to division. You can write these operators using vector and matrix names. As an example, consider the cross product of a matrix and a vector:
We'll look at this operation in more detail later. Note, however, that it requires nine multiplications and nine additions. Writing them is tedious and error-prone - it is very difficult to do this for large matrices.
The use of vectors is very wide. Think of the pixels on a monitor screen - there can be millions of them. They are processed using matrix operations.
INMathcad
To create a vector or matrix, open the Matrices / Tables tab. When the cursor is in an empty area, click on the leftmost "Insert Matrix" button. A grid with small squares will appear:
Move the pointer over the grid, select the desired matrix size, then left-click. An empty matrix will appear:
A matrix can be given a name by clicking on the left parenthesis, pressing [:] for the assignment operator and entering a name:
Inserting and deleting rows and columns is easy using commands from the Vector / Matrix Operators menu on the Matrices and Tables tab:
Matrix operations
The effect of various operations on matrices and vectors will be easier to understand using symbols. We will use two matrices and two vectors:
Transpose
The transposition operator is located on the Mathematics -\u003e Operators -\u003e Vectors and Matrices tab:
Click on the right border of the matrix and apply the operator. It works for both character and numeric matrices:
Item-by-item operations
Often operations in vectors have to be performed element by element. The vectorization operator is used for this. Operations in Excel are often element-wise, and they are also important in Mathcad. To multiply two vectors element by element, first enter a simple multiplication:
Then select the whole expression and apply vectorization:
Calculate to see the result: the first element is multiplied by the first, the second by the second, etc .:
Other element-wise operations:
Element-wise operations only apply to arrays of the same size.
Addition and subtraction
Addition and subtraction is performed element by element:
This operation also applies only to arrays of the same size.
Using the sum operator, you can find the sum of all elements of a vector (not a matrix):
Scalar product
Multiplication by a constant works like this:
In scalar matrix multiplication, rows are multiplied by columns. This uses the same symbol as for normal multiplication. This operation is valid only for those matrices in which the number of rows in the first matrix is \u200b\u200bequal to the number of columns in the second. For our 2x2 matrices:
Note that the sequence of factors plays a role:
The dot product is not commutative except in special cases:
The scalar product of two vectors gives the result with complex conjugate numbers (with a bar above). For real numbers, you can ignore this:
Vector product
This operator is applicable only to two column vectors with three elements:
The vector product is widely used in mechanics, hydrodynamics, electromagnetism, and other fields.
inverse matrix
The inverse matrix is \u200b\u200bdefined only for square matrices.