The determinant is an algebraic operation that transforms a square matrix into a scalar. This operation has many useful and important properties. For example, the determinant is zero if and only the matrix is singular (no inverse exists). The determinant also has an important geometric interpretation as the area of a parallelogram, and more generally as the volume of a higher-dimensional parallelepiped.
The notion of determinant predates matrices and linear transformations. Originally, the determinant was a number associated to a system of linear equations in variables. This number “determined” whether the system possessed a unique solution. In this sense, two-by-two determinants were considered by Cardano at the end of the 16th century and ones of arbitrary size (see the definition below) by Leibniz about 100 years later.
Let be an matrix with entries that are elements of a given field11Most scientific and geometric applications deal with matrices made up of real or complex numbers. However, the determinant of a matrix over any field is well defined sense and has all the properties of the more conventional determinant. Indeed, many properties of the determinant remain valid for matrices with entries in a commutative ring.. The determinant of , or for short, is the scalar quantity
The index in the above sum varies over all the permutations of (i.e., the elements of the symmetric group .) Hence, there are terms in the defining sum of the determinant. The symbol denotes the parity of the permutation; it is according to whether is an even or odd permutation. Using the Einstein summation convention one can also express the above definition as
where we’ve raised the first index so that , and where
is known as the Levi-Civita permutation symbol.
By way of example, the determinant of a matrix is given by
There are six permutations of the numbers , namely
the overset sign indicates the permutation’s signature. Accordingly, the deterimant is a sum of the following terms:
Remarks and important properties
In particular, if we let be the matrix representing a change of basis, this shows that the determinant is independent of the basis. The same is true of the trace of a matrix. In fact, the whole characteristic polynomial of an endomorphism is definable without using a basis or a matrix, and it turns out that the determinant and trace are two of its coefficients.
The determinant of a matrix is zero if and only if is singular; that is, if there exists a non-trivial solution to the homogeneous equation
The transpose operation does not change the determinant:
The determinant is homogeneous of degree . This means that
|Date of creation||2013-03-22 12:33:07|
|Last modified on||2013-03-22 12:33:07|
|Last modified by||rmilson (146)|