Matrix Algebra - The basic definitions
The basic definitions
System from mn numbers (real, complex), either functions, or other objects, recorded in the form of the rectangular table consisting from m rows and n columns:
is called matrix.
Numbers (functions, other objects) , making the matrix (1), are called elements of a matrix. Here the first index i designates the row number, and the second j – the column number on intersection of which the given element of a matrix is located.
For a matrix (1) there is a shorthand record:
or simply . In this case speak, that the matrix A has dimension m×n. If m=n the matrix is called square of the order n. If m≠n the matrix is called rectangular. The matrix of dimension 1×n is called a vector-row, and a matrix of dimension m×1 – a vector-column. It is possible to consider usual number (scalar) as a matrix of dimension 1 × 1.
If the square matrix looks like:
then it is called diagonal matrix.
If in the diagonal matrix (2) all diagonal elements are equal to1 then this matrix is called identity matrix and is designates as:
Using Kronecker’s symbol
it is possible to record:
The matrix at which all elements are equal to 0 is called zero matrix and is designated 0.
Elements of the n-th order square matrixform a so-called main diagonal of a matrix.
The sum of elements of the main diagonal is called a trace or a spur of a matrix:
The concept determinant is connected with a square numerical matrix:
Matrix and its determinant different (though and connected) concepts. The numerical matrix A is the ordered system of numbers recorded in the form of the rectangular table, and its determinant det A is the number equal:
where the sum (4) extends on possible permutations of elements 1, 2..., n and, consequently, contains n! addends, and k = 0, if the permutation is even and k = 1, if the permutation is odd.
E x a m p l e . Calculate a determinant of a matrix
S o l u t i o n . According to (3) we have: