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Contents >> Applied Mathematics >> Matrix Algebra >> Principles of Matrix Calculation >> The basic definitions

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 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:

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Last updated: April 30, 2015.