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02-10-03 - Lecture 5 Andrei Antonenko 1 Matrices Now well...

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Lecture 5 Andrei Antonenko February 10, 2003 1 Matrices Now we’ll start studying new algebraic object — matrices. Definition 1.1. The matrix is a (rectangular) table of the elements of R . (Actually, we can consider matrices over fields other than R — in the future we will work with matrices over the field of complex numbers C .) Now we’ll introduce some notation that we will use. We will denote matrices with capital letters, and the elements of the matrix with same small letter with 2 subscripts, the first of them denotes the row, and the second one denotes the column. Often we will speak about m × n -matrices, which means that it has m rows and n columns. A = ( a ij ) , A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . . . . a m 1 a m 2 · · · a mn The matrix is called square matrix if the number of its rows is equal to the number of its columns. For every square matrix we will define its main diagonal , or simply diagonal , as a diagonal from the top left corner to the bottom right corner, i.e. diagonal consists of the elements a 11 , a 22 , . . . , a nn . Another diagonal is called secondary . It is used very rarely. So, we introduced an object. But now we should introduce operations, otherwise the object is not interesting! 2 Matrix Operations 2.1 Addition The first and the easiest matrix operation is matrix addition. 1
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Definition 2.1. Let A and B are m × n -matrices. Then their sum C = A + B is an m × n - matrix such that c ij = a ij + b ij , i.e. the elements of this matrix are sums of corresponding
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