The definition, that I have learned, of the determinant of a matrix is a set of instructions on how to calculate its value. This definition can be put in equation form if we keep track of odd permutations and even permutations of products of matrix elements but this is a set of instructions appended to the equation. Is there an equation for the determinant of an nxn matrix that is self contained in the sense of not requiring that it be accompanied by an appended set of instructions?
Perhaps Leibniz or Laplace came close to answer your question? Please see:
en.wikipedia.org/wiki/Leibniz_formula_for_determinants
en.wikipedia.org/wiki/Laplace_expansion
Perhaps Leibniz or Laplace came close to answer your question? Please see:
en.wikipedia.org/wiki/Leibniz_formula_for_determinants
en.wikipedia.org/wiki/Laplace_expansion
The following link may be useful,
http://home.iitk.ac.in/~rksr/html/02MAT2.htm
Hi Larry:
It ends up being the Leibniz formula for determinants, but it is all about volume, and negative volume thanks to the (sgn) instruction keeping track of the parity. And this all comes from three ideas about what we expect volume to be like:
a) the volume of a unit cube is one
b) if the parallelepiped is degenerate then the volume is zero
c) volume scales with the length of its sides and shears do not change Euclidean volumes.
This is all written up in my new favorite math book on pages 16 to 22, A Visual Introduction to Differential Forms and Calculus on Manifolds by Jon Pierre Fortney
Thank you George Stoica, Adiya, and George Soli. You all gave good answers to my question. The specific reason that I wanted this information (which I did not explain in my question but involves Jacobians and volume integrals) makes George Soli's answer the perfect answer for my needs.
You may check
Equation for the direct determination of the density matrix: Time-dependent density equation and perturbation theory
Hiroshi NakatsujiJournal:Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta)Year:1999
Dear L.D. Edmonds
In addition to the helpful answers, I would like to add,
we can calculate the determinant of a square matrix A by using the trace of A and its powers.
Examples:
n=2 , detA = 1/2[( trA)2 - (trA2 )]
n=3, detA=1/6[ (trA)3 - 3( trA)(trA2 ) + 2( trA3)]
A general formula is available.
Best regards
Dear Auwal, I was looking for an equation that expresses the determinant in terms of elementary arithmetic (addition, multiplication, etc.) operations. The usual textbook equation requires that we determine whether a permutation of numbers is odd or even. Determining whether a permutation is odd or even does not seem, to me, to be an elementary operation. In contrast, the trace of a matrix can be expressed in terms of elementary operations. This makes the answer given by Issam a good answer.
L.D. Edmonds
Yes the trace of a matrix can be expressed in terms of elementary operations.
But calculate A*A , (or A*A*A) is not easy and required n^3 operations (or more).
@ Mohamed Azzedine
Read the question carefully.
The question is not about the number of steps! (complexity).
It is about the reduction of instructions!
The standard Laplace or Leibniz formulae, see e.g.
https://en.wikipedia.org/wiki/Determinant
do not provide a computationally viable approach for matrices that have even modest size. In numerical linear algebra, the standard procedure is to compute the LU factorization of the matrix and then take the product of the elements on the diagonal of the upper triangular U matrix, see e.g. the classical book
Golub and Van Loan, Matrix computation, Johns Hopkins University Press.
While this may seem complicated, it is a procedure that only entails elementary arithmetic operations and it has the advantage of being applicable also to very large matrices. On the other hand, it is well known that for large matrices the determinant not such a useful tool to determine the matrix properties, since a matrix can have vanishingly small determinant even when it is very far from being singular or difficult to invert numerically.
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