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Topic: Determinant

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  Determinant - Wikipedia, the free encyclopedia
Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule.
Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors.
The Pfaffian is an analog of the determinant for
en.wikipedia.org /wiki/Determinant   (2284 words)

 DETERMINANT - LoveToKnow Article on DETERMINANT   (Site not responding. Last check: 2007-10-22)
Observe that for a determinant of the n -th order, taking the decomposition to be i + (n 1), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant.
The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations.
Determinants composed of binomial coefficients have been studied by V. von Zeipel; the expression of definite integrals as determinants by A. Tissot and A. Enneper, and the expression of continued fractions as determinants by Jacobi, V. Nachreiner, S. Gunther and E. FUrstenau.
13.1911encyclopedia.org /D/DE/DETERMINANT.htm   (1578 words)

 PlanetMath: determinant
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 determinant operation is multi-linear, and anti-symmetric with respect to the matrix's rows and columns.
The determinant of a lower triangular, or an upper triangular matrix is the product of the diagonal entries, since all the other summands in (1) are zero.
planetmath.org /encyclopedia/Determinant2.html   (490 words)

 Slater determinant - Wikipedia, the free encyclopedia
A Slater determinant (named after the physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many-fermion system, which by construction satisfies the Pauli principle.
In the same way, the use of Slater determinants assures the obeying of the Pauli principle; the determinant will vanish if any of the two spin-orbitals are identical, for this leads to two identical columns.
A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree-Fock theory.
en.wikipedia.org /wiki/Slater_determinant   (361 words)

 Determinant Notes   (Site not responding. Last check: 2007-10-22)
Determinant notes in PDF form or the same in larger font.
Determinants are motived geometrically and a simple set of axioms based on row operations are used.
Determinants are used in the substitution formula for multiple integrals, and the geometric approach taken has obvious utility there.
www.cs.uleth.ca /~holzmann/notes/det.html   (226 words)

 Determinant: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-10-22)
Determinants are used to characterize invertible matrices[Click link for more facts about this topic] (namely as those matrices, EHandler: no quick summary.
is times the determinant of the matrix that results from by removing the -th row and the -th column.
In vector calculus, the jacobian is shorthand for either the jacobian matrix or its determinant, the jacobian determinant....
www.absoluteastronomy.com /encyclopedia/d/de/determinant.htm   (4169 words)

 6.4 - The Determinant of a Square Matrix
The minor is the determinant that remains when you delete the row and column of the element you're trying to find the minor for.
The inverse of a matrix is found by dividing the adjoint of the matrix by the determinant of the matrix.
With determinants, since the determinant of a transpose is the same as the determinant of the matrix, the elementary row operations can also be applied to columns.
www.richland.edu /james/lecture/m116/matrices/determinant.html   (2597 words)

 AllRefer.com - determinant (Mathematics) - Encyclopedia
determinant, a polynomial expression that is inherent in the entries of a square matrix.
of an nth-order determinant is the determinant of order (n-1) formed by deleting the ith row and the jth column of the original determinant.
In expanding a determinant by minors, first the minor of every element in a particular row or column is formed.
reference.allrefer.com /encyclopedia/D/determinnt.html   (430 words)

 determinant on Encyclopedia.com
Cholesteryl ester transfer protein gene polymorphism is a determinant of HDL cholesterol and of the...
DETERMINANT [determinant] a polynomial expression that is inherent in the entries of a square matrix.
of an n th-order determinant is the determinant of order (n -1) formed by deleting the i th row and the j th column of the original determinant.
www.encyclopedia.com /html/d1/determinnt.asp   (654 words)

In mathematics, an array of elements written as a square, and denoted by two vertical lines enclosing the array.
Determinants are used to solve sets of simultaneous equations by matrix methods.
When applied to transformational geometry, the determinant of a 2 × 2 matrix signifies the ratio of the area of the transformed shape to the original and its sign (plus or minus) denotes whether the image is direct (the same way round) or indirect (a mirror image).
www.tiscali.co.uk /reference/encyclopaedia/hutchinson/m0006669.html   (293 words)

 Determinant   (Site not responding. Last check: 2007-10-22)
To find the determinant of larger and larger matrices, you would need to contend with faster-than-exponential growth of the number of permutations, if you want to carry out the calculations as I have described thus far.
Thus, even though AB BA, because the commutative property doesn't hold for matrix multiplication, the determinants of these two matrices (AB and BA) are the same, and they are the product of the determinants of the two factor matrices.
Then, since the determinant is "preserved" by matrix multiplication, and AB = I, it follows that ab=1.
mcraefamily.com /MathHelp/MatrixDeterminant.htm   (555 words)

 Determinant   (Site not responding. Last check: 2007-10-22)
A determinant is a square array of elements with a prescribed evaluation rule which is depicted as follows
Determinants are useful in the evaluation of vector products and vector operations like the curl.
The evaluation of the determinant of the cofactor follows the same pattern until the cofactor has dimension two.
hyperphysics.phy-astr.gsu.edu /hbase/deter.html   (120 words)

 Updating the Slater Determinant
Fortunately, there is a method for updating the wavefunction[115, 26] which is much less costly than recalculating the whole determinant every time an electron is moved.
The quantity that is calculated is the ratio of the new and old determinants.
The cost of updating the quantities involved in the determinant, if the move is accepted, is greatly reduced by making use of this ratio.
www.tcm.phy.cam.ac.uk /~ajw29/thesis/node130.html   (346 words)

 DETERMINANT - Online Information article about DETERMINANT
rule for the derivation of the determinants of the orders r, 2, 3, 4, andc., each from the preceding one, viz.
article, consider the determinant ax +by +cz -d b,c ; a'x+b'y+c'z-d', b', c' a"x+b"y+c"z—d", b", c" it appears that this is =xla,b,c I+y b,b,cI+zlc,b-,c,d,b,c I; r r ' ' r '.
Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant.
encyclopedia.jrank.org /DEM_DIO/DETERMINANT.html   (1971 words)

determinant, a polynomial expression that is inherent in the entries of a square
of the square matrix, as determined from the number of entries in any row or column, is called the order of the determinant.
determinism - determinism, philosophical thesis that every event is the inevitable result of antecedent causes.
www.factmonster.com /ce6/sci/A0815307.html   (380 words)

 Definition of Determinant
Determinant is a function which as an input accepts
There are easier ways to compute the determinant rather than using this formula.
There are other ways of computing the determinant of a given matrix, for example row operations can be used to reduce the matrix to a triangular matrix whose determinant is the product of the diagonal entries, another way is using cofactor expansion.
www.math.ucdavis.edu /~daddel/linear_algebra_appl/Applications/Determinant/Determinant/node3.html   (186 words) Determinant and Eigenstructure   (Site not responding. Last check: 2007-10-22)
As is the case of inversion of a square matrix, calculation of the determinant is tedious and computer assistance is needed for practical calculations.
If the determinant of the (square) matrix is exactly zero, the matrix is said to be singular and it has no inverse.
Of great interest in statistics is the determinant of a square symmetric matrix D whose diagonal elements are sample variances and whose off-diagonal elements are sample covariances.
www.itl.nist.gov /div898/handbook/pmc/section5/pmc532.htm   (420 words)

De determinant of the matrix is the element itself.
This is, by the definition of a determinant, the determinant of the sub-matrix of A obtained from A by crossing out the first row and the first column.
Now the cofactor of e is the determinant of the sub-matrix of obtained from by crossing out the first row and the first column.
www.ping.be /~ping1339/determ.htm   (1885 words)

 Math.com Online Solvers Matrices Determinant Help   (Site not responding. Last check: 2007-10-22)
The determinant command in the matrices section of Quickmath allows you to find the determinant of any square matrix (a matrix which has the same number of rows and columns).
The input form of a solution can be copied and pasted into a matrix input field, so you can use the result from one calculation in another calculation without having to re-type it.
Here are some examples illustrating the types of matrices you can use the determinant command on and the results which QuickMath will return.
www.math.com /students/solvers/matrices/determinant_help.htm   (207 words)

The determinants of a matrix and its transpose are equal,
Multiplying all the elements of the row (column) of the determinant by the same number the determinant will be multiplied by the same number.
The determinant will not change if add to the row (column) an arbitrary row (column) multiplied by an arbitrary number.
www.cs.ut.ee /~toomas_l/linalg/lin1/node14.html   (638 words)

The former is the determining, the latter the determinable principle.
These elements determine not only the natural warmth or moisture of bodies, but also their natural motion, upward or downward, according to the preponderance of air or earth.
Aristotle answers that man's happiness is determined by the end or purpose of his existence, or in other words, that his happiness consists in the "good proper to his rational nature".
www.newadvent.org /cathen/01713a.htm   (5735 words)

 Cramer's Rule and the Determinant
Since the determinant is defined as the sum of products, if we could introduce(legally) some zeros into the matrix, forcing some of those products to be zero then our job would be much easier.
It turns out that the value of the determinant is equal to the product of the main diagonal when the matrix is in upper triangular form.
The value of the specific unknown is determined by dividing the special determinant by the main determinant.
www.aspire.cs.uah.edu /textbook/deter.html   (775 words)

 Search Results for Determinant
These were important in that for the first time the definition of the determinant was made in an algorithmic way and the entries in the determinant were not specified so his results applied equally well to cases were the entries were numbers or to where they were functions.
An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note On determinant theory.
This was done by searching for the sums or the expression of their general terms and to those of the roots or determinant equations, by which to obtain with a simple calculation the approximate value of the products or the indefinite sums of certain numbers.
www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Determinant&CONTEXT=1   (1253 words)

 Determinant Word Problems
Remember that a determinant can be negative, while areas (and absolute values) cannot.
The order of the points in the determinant doesn't matter because changing the order of the rows in a determinant doesn't change its value, except maybe for changing the sign.
This last problem shows why you will need to be able to evaluate determinants by hand: your teacher can give you a problem like this that you can't solve with your calculator.
www.purplemath.com /modules/detprobs.htm   (419 words)

 mkaz.com : Java Matrix Calculator
The calculation of the determinant, by definition, is based upon a factorial number of calculations with respect to the size of the matrix.
The determinant of A-inverse equals 1 over the determinant of A. If two rows of matrix A are equal, the determinant of A equals 0.
The determinant of A-transpose equals the determinant of A. If the matrix B is constructed by interchanging two rows (columns) in matrix A, then the determinant of B equals the negative determinant of A
www.mkaz.com /math/matrix.html   (477 words)

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