Matrix. A matrix, here denoted by a boldface capital letter, is a rectan­gular array of real numbers arranged as follows:

A matrix such as A in (11.1.1), which has n rows and m columns, is called an n X m (read “n by m”) matrix. Matrix A may also be denoted by the symbol {fly}, indicating that its i, jth element (the element in the ith row and jth column) is aly

Transpose. Let A be as in (11.1.1). Then the transpose of A, denoted by A’, is defined as an и X и matrix whose i, jth element is equal to a]r For example,

1 Подпись: 1 2 3 4 5 64 ‘

2 5

3 6

Note that the transpose of a matrix is obtained by rewriting its columns as rows.

Square matrix. A matrix which has the same number of rows and col­umns is called a square matrix. Thus, A in (11.1.1) is a square matrix if

n = m.

Symmetric matrix. If a square matrix A is the same as its transpose, A is called a symmetric matrix. In other words, a square matrix A is symmetric if A’ = A. For example,

1 4 6 4 2 5 6 5 3

is a symmetric matrix.

Vector. An n X 1 matrix is called an n-component column vector, and a 1 X n matrix is called an n-component row vector. (A vector will be denoted by a boldface lowercase letter.) If b is a column vector, b’ (trans­pose of b) is a row vector. Normally, a vector with a prime (transpose sign) means a row vector and a vector without a prime signifies a column vector.

Diagonal matrix. Let A be as in (11.1.1) and suppose that n = m (square matrix). Elements au, a^, . . . , ann are called diagonal elements. The other elements are off-diagonal elements. A square matrix whose off-di­agonal elements are all zero is called a diagonal matrix.

Identity matrix. An n X n diagonal matrix whose diagonal elements are all ones is called the identity matrix of size n and is denoted by In. Sometimes it is more simply written as I, if the size of the matrix is apparent from the context.

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