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Matrix Determinant

In addition to the rank of a matrix, the determinant is also an important characteristic number of a matrix. The determinant of a matrix A is depicted as |A|. In order to understand the definition of a determinant, we first have to introduce the term permutation:
 
 
Permutation Let (n1, n2, n3, ..., nk) be an ordered set of k arbitrary numbers. Then any ordered arrangement of the same numbers, say, (np1, np2, np3, ..., npk) is called a permutation of that set, in our case denoted by p.
Odd (Even) Permutation We call every case in which both ps precedes pr, and ps is greater than pr in a permutation (p1, p2, p3, ..., pk) of (1, 2, 3, ..., k) an inversion of this permutation p. p is called odd or even, depending on whether the number of inversions is even or odd.

 

Example: an even permutation
(3, 1, 4, 2) is a permutation of (1, 2, 3, 4). The 3 in this permutation has 2 inversions: It precedes both 1 and 2, while being larger than 1 and 2. It also precedes 4, but is smaller than 4, so this doesn't count. 1 has no inversions, neither has 4, nor 2. We thus have 2+0+0+0 = 2 inversions, so the permutation (3, 1, 4, 2) is even.

 

After these definitions concerning permutations, we can define the determinant of a matrix as follows: 
 
Determinant of a Matrix Let A be an arbitrary square matrix. Then its determinant (denoted with |A|) is the sum over the product of all permutations of all elements within any row, multiplied by either +1 or -1, depending on whether the respective permutation is even or odd.

 

Note that determinants are only defined for square matrices, and that the determinant of a square matrix and its transposed are equal. There is also an important relationship regarding the product of (square) matrices and their determinants:

 

Product of Determinants Let A, B, and C be square matrices of the same order. If C=AB, then |C|=|A||B|.

The meaning of the determinant

An important rule about determinants is that |A| is always 0 if, and only if, at least two vectors of a matrix are linear-dependent. However in this case, the rank of a matrix is usually a better measure since it additionally provides information on how many vectors are dependent. In practical situations, however, the vectors of a matrix may be linearly independent in a formal sense, but may be very similar to each other, resulting in a near-dependence. Such situations may result in numerical instabilities and must be detected.

In general, the determinant increases with increasing independence of the vectors constituing the matrix. In fact, the determinant defines the volume of the geometrical shape which is spanned by the vectors of a matrix. The more the vectors become "similar" (that means they point in the same direction), the smaller the volume is.

For non-square matrices, it can be shown that there are always some vectors (either rows or columns) which are not independent of the other vectors. Therefore we can simply define the determinant of such a non-square matrix as always zero.


Last Update: 2005-Jšn-25