James Joseph Sylvester (1814-1897) England
James Joseph Sylvester (1814-1897) England
James Joseph Sylvester made fundamental contributions to matrix theory, invariant theory, number theory and combinatorics. He determined all matrices that commute with a given matrix. He introduced many mathematical terms including “discriminant”.
In 1880, the Royal Society of London awarded Sylvester the Copley Medal, a highest award for scientific achievement. In 1901, Royal Society of London instituted the Sylvester medal in his memory, to encourage mathematical research.
In this chapter we are going to discuss an important mathematical object called “MATRIX”. Here, we shall introduce matrices and study the basics of matrix algebra. Matrices were formulated and developed as a concept during 18th and 19th centuries. In the beginning, their development was due to transformation of geometric objects and solution of linear equations. However matrices are now one of the most powerful tools in mathematics. Matrices are useful because they enable us to consider an array of many numbers as a single object and perform calculations with these symbols in a very compact form. The “ mathematical shorthand” thus obtained is very elegant and powerful and is suitable for various practical problems. The term “Matrix” for arrangement of numbers, was introduced in 1850 by James Joseph Sylvester. “Matrix” is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced. Now let us consider the following system of linear
equations in x and y :
3 2 4 x y - = (1)
2 5 9 x y + = (2)
We already know how to get the solution (2, 1) of this system by the method of elimination (also known as Gaussian Elimination method), where only the coefficients are used and not the variables. The same method can easily be executed and the solution can thus be obtained using matrix algebra.
A matrix is a rectangular array of numbers in rows and columns enclosed within
square brackets or parenthesis.
A matrix is usually denoted by a single capital letter like A, B, X, Y,g . The
numbers that make up a matrix are called entries or elements of the matrix. Each horizontal
arrangement in a matrix is called a row of that matrix. Each vertical arrangement in a
matrix is called a column of that matrix.
Some examples of matrices are
A , 1
4
2
5
3
6 = c m B
2
3
1
0
8
5
1
9
1
= -
-
-
> H and C
1
0
1
= f p
In a m n # matrix, the first letter m always denotes the number of rows and the second letter n always denotes the number of columns. A unit matrix is also called an identity matrix with respect to multiplication. Every unit matrix is clearly a scalar matrix. However a scalar matrix need not be a unit matrix. A unit matrix plays the role of the number 1 in numbers.
(i) A zero-matrix need not be a square matrix. (ii) Zero-matrix plays the role of the
number zero in numbers. (iii) A matrix does not change if the zero-matrix of same
order is added to it or subtracted from it. Multiplication of two diagonal matrices of same order is commutative. Also, under matrix multiplication unit matrix commutes with any square matrix of same order.
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