释义 |
DeterminantDeterminants are mathematical objects which are very useful in the analysis and solution of systems of linear equations. Asshown in Cramer's Rule, a nonhomogeneous system of linear equations has a nontrivial solution Iff thedeterminant of the system's Matrix is Nonzero (so that the Matrix is nonsingular). A determinant is defined to be
 | (1) |
A determinant can be expanded by Minors to obtain | |  | (2) | A general determinant for a Matrix A has a value
 | (3) |
with no implied summation over and where is the Cofactor of defined by
 | (4) |
Here, C is the Matrix formed by eliminating row and column from A, i.e., by Determinant Expansion by Minors.
Given an determinant, the additive inverse is
 | (5) |
Determinants are also Distributive, so
 | (6) |
This means that the determinant of a Matrix Inverse can be found as follows:
 | (7) |
where I is the Identity Matrix, so
 | (8) |
Determinants are Multilinear in rows and columns, since
 | (9) |
and
 | (10) |
The determinant of the Similarity Transformation of a matrix is equal to the determinant of the original Matrix
 | (11) |
The determinant of a similarity transformation minus a multiple of the unit Matrix is given by
The determinant of a Matrix Transpose equals the determinant of the original Matrix,
 | (13) |
and the determinant of a Complex Conjugate is equal to the Complex Conjugate of the determinant
 | (14) |
Let be a small number. Then
 | (15) |
where is the Trace of A. The determinant takes on a particularly simple form for aTriangular Matrix
 | (16) |
Important properties of the determinant include the following. - 1. Switching two rows or columns changes the sign.
- 2. Scalars can be factored out from rows and columns.
- 3. Multiples of rows and columns can be added together without changing the determinant's value.
- 4. Scalar multiplication of a row by a constant
multiplies the determinant by . - 5. A determinant with a row or column of zeros has value 0.
- 6. Any determinant with two rows or columns equal has value 0.
Property 1 can be established by induction. For a Matrix, the determinant is
For a Matrix, the determinant isProperty 2 follows likewise. For and matrices,
 | (19) |
and
 | (20) |
Property 3 follows from the identity | |  | (21) |
If is an Matrix with Real Numbers, then has the interpretation as the oriented -dimensional Content of the Parallelepiped spannedby the column vectors , ..., in . Here, ``oriented'' means that, up to a change of or Sign, the number is the -dimensional Content, but the Sign depends on the ``orientation'' ofthe column vectors involved. If they agree with the standard orientation, there is a Sign; if not, there is a Sign. The Parallelepiped spanned by the -D vectors through is the collectionof points
 | (22) |
where is a Real Number in the Closed Interval [0,1].
There are an infinite number of determinants with no 0 or entries having unity determinant. Oneparametric family is
 | (23) |
Specific examples having small entries include
 | (24) |
(Guy 1989, 1994).See also Circulant Determinant, Cofactor, Hessian Determinant, Hyperdeterminant, Immanant,Jacobian, Knot Determinant, Matrix, Minor, Permanent, Vandermonde Determinant,Wronskian References
Arfken, G. ``Determinants.'' §4.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 168-176, 1985.Guy, R. K. ``Unsolved Problems Come of Age.'' Amer. Math. Monthly 96, 903-909, 1989. Guy, R. K. ``A Determinant of Value One.'' §F28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 265-266, 1994.
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