Is A×B necessarily equal to B×A?
In social production practice and scientific research, some related data, such as experimental data, statistical data, financial data, etc., are often processed. In order to clearly display these data, people usually make a table, which can be abstracted into a matrix form: a rectangular array of n×m numbers arranged in n rows and m columns is called an n×m matrix, enclosed in brackets, each number is an element of the matrix. For example, the sales volume of each store of a commercial chain company can correspond to a 2×3 matrix.
In the middle of the 19th century, the British mathematician Cayley systematically established matrix theory and stipulated the arithmetic operations of matrices. Matrix addition is relatively simple, two matrices with the same number of rows and columns, and the matrix obtained by adding elements at the corresponding positions. The regulations for matrix multiplication are different: when two matrices are multiplied, the number of columns in the previous matrix is required to be equal to the number of rows in the latter matrix, and the elements in the ith and jth columns of the product are equal to the number obtained by multiplying and summing the corresponding position elements in the ith row of the previous matrix and the jth column of the latter matrix.
Some beginners to matrices don't understand matrix multiplication very well. Why is its rule so odd instead of multiplying corresponding positional elements like addition? In fact, defining matrix multiplication in this way is more in line with practical needs. Taking the data of the above-mentioned commercial company as an example, the calculation of turnover and profits of multiple stores and multiple commodities can be used to correspond to the relationship between tables through matrix multiplication, which is the need for actual calculation; more importantly, the need for linear transformation in mathematics. After the relationship between variables is substituted, the coefficient matrix satisfies the multiplication relationship, which shows that this definition is very natural.
Matrix multiplication also has a strange property. As we all know, when two numbers a and b are multiplied, there is always a×b=b×a, which is the commutative law of multiplication. However, for matrix multiplication, if A and B represent two matrices, usually A×B and B×A are not equal. The calculation result is related to the order of the two matrices multiplied, which is quite different from common multiplication operations.
People have become accustomed to the well-known commutative law of multiplication and have doubts about the non-commutative nature of matrix multiplication. However, this non-commutative matrix multiplication has found its use in the creation of quantum mechanics. In the summer of 1925, the 24-year-old German physicist Heisenberg constructed a new set of quantum theory. The multiplication results used depend on the order of multiplication. He handed the paper to his mentor Born. Born thought of matrix multiplication. The multiplication used by Heisenberg was matrix multiplication, which was unfamiliar to most physicists at that time. Later, this theory was called matrix mechanics and an important part of quantum mechanics. Heisenberg won the Nobel Prize in Physics in 1932 for his important contribution to the creation of the theory of quantum mechanics.