Matrix calculator

In mathematics, to compactly write systems of linear equations, matrices are often used, written in the form of rectangular tables. In these tables, the number of rows corresponds to the number of equations, and the number of columns corresponds to the number of unknowns. There are also matrices in the form of rings and fields: for writing complex and real numbers.

With the help of matrix tables, you can solve algebraic and differential equations, reducing calculations to operations on matrices, which greatly speeds up the process. In addition, it simplifies the systematization of large data arrays, including those in electronic computing devices.

History of occurrence

Historians attribute the invention of the first matrices to the ancient Chinese. More than 4000 years ago, during the reign of Emperor Yu the Great, these mathematical objects were called magic squares, and allowed complex calculations to be carried out in a few simple steps.

According to ancient Chinese legend, the first magic square with hieroglyphs was discovered on the shell of a sacred tortoise that surfaced from the Yellow River in 2200 BC. The matrix found application in trade and engineering, and subsequently spread to many countries of the Ancient East. During the early Middle Ages, they learned about it in the Arab countries, in the 11th century - in India, in the 15th-16th centuries - in Japan.

In Europe, the magic square was known only at the turn of the 15th-16th centuries - thanks to the Byzantine writer Manuel Moskhopul, who described it in his writings. In 1514, the German painter Albrecht Dürer included a magic square in his engraving "Melancholia". On it, among other objects, a square is depicted, in the central cells of which the date of the creation of the engraving is inscribed.

In the 16th century, numerical matrices became widespread among soothsayers and astrologers, who gave the magic square mystical and healing properties. It can often be found on miniature silver engravings of the time, which supposedly protected their owners from the plague. Then, in the 16th century, practical applications were found for matrices in Europe. The German philosopher Cornelius Heinrich Agrippa used them to describe the motion of the 7 planets by constructing squares from the 3rd to the 9th order.

In the 17th and 18th centuries, research continued, and in 1751 the Swiss mathematician Gabriel Cramer published a new way of solving algebraic equations using matrices with zero principal determinant, which he had been working on for several decades.

At about the same time, the Gauss method for solving a system of linear algebraic equations was published. Although today its name is inextricably associated with the name of a German mathematician, the authorship, according to historians, does not belong to him. So, this method of calculating matrices was known 2000 years before the life of Carl Friedrich Gauss, and was presented in the ancient Chinese “Mathematics in Nine Books” in the 2nd century BC.

As algebra and operational calculus developed, interest in matrices flared up with renewed vigor in the 19th and 20th centuries. Outstanding scientists of their time were engaged in their research: William Hamilton, Arthur Cayley and James Joseph Sylvester.

By the middle of the 19th century, they finally formulated the rules for adding and multiplying matrix tables, and by the beginning of the 20th century, the theoretical base was expanded by the studies of Karl Weierstrass and Ferdinand Georg Frobenius. It is noteworthy that the matrix received its modern name and designation only in 1841 - thanks to the English mathematician Arthur Cayley.

Varieties of matrices

A standard rectangular matrix is a number series with m number of rows and n number of columns. All elements in it are numbered from left to right and from top to bottom. The top row can be represented as (a₁ a₂ a₃ ... aₙ) and the bottom row as (aₘ₁ aₘ₂ aₘ₃ ... aₘₙ). The matrix size is specified as m × n, where m and n are natural numbers.

Accordingly, to find out the total number of elements in the table, it is enough to multiply m by n: the number of rows by the number of columns. What other matrices exist besides rectangular?

• Square. They have the same number of rows and columns, that is, m = n.
• As a column vector. Such a matrix has n = 1, and the size is specified as "m × 1". All numbers in it are numbered from top to bottom: colon (a₁ a₂ ... aₘ).
• As a row vector. A matrix similar to the previous one, but with m = 1 and size "1 × n". The numbers in it are numbered from left to right: row (a₁ a₂ ... aₙ).

Columns and rows are denoted by capital letters (m, n), but in general terms, each matrix can be represented as K = M × N, even if one of the values \u200b\u200bis equal to one.

There are also transposed, diagonal, identity and zero matrices. In the identity matrix, all elements are units; when multiplied by it, any matrix remains unchanged. In zero, all rows and columns consist of zeros, each matrix remains unchanged when added to it.