defining determinant

[AntonioLao]

The determinant (det) of a given square matrix is simply the sum of the products of element permutations. The signs of these product addends can be positive or negative depending whether the product of differences of column indices are positive or negative.

For example, the det of a 2 by 2 matrix with elements a11, a12, a21, a22 has two product terms a11a22 and a12a21. The first is positive since 2-1=1. The second is negative since 1-2=-1. Since the factorial of 2 is 2 all 2 by 2 matrices have two product addends. For a 6 by 6 matrix, the number of product addends is 6!=720. Half are positives and half are negatives. One of these could have column element permutations as a13a25a31a44a56a62 and the product of column index differences (5-3)(1-3)(4-3)(6-3)(2-3)(1-5)(4-5)(6-5)(2-5)(4-1)(6-1)(2-1)(6-4)(2-4)(2-6) is negative therefore the term is negative. On the other hand if the column element permutation is a11a22a33a44a55a66 then the product of index differences (2-1)(3-1)(4-1)(5-1)(6-1)(3-2)(4-2)(5-2)(6-2)(4-3)(5-3)(6-3)(5-4)(6-4)(6-5) is positive.

[mkirkpatrick]

The determinant (det) of a given square matrix is simply the sum of the products of element permutations. The signs of these product addends can be positive or negative depending whether the product of differences of column indices are positive or negative.

For example, the det of a 2 by 2 matrix with elements a11, a12, a21, a22 has two product terms a11a22 and a12a21. The first is positive since 2-1=1. The second is negative since 1-2=-1. Since the factorial of 2 is 2 all 2 by 2 matrices have two product addends. For a 6 by 6 matrix, the number of product addends is 6!=720. Half are positives and half are negatives. One of these could have column element permutations as a13a25a31a44a56a62 and the product of column index differences (5-3)(1-3)(4-3)(6-3)(2-3)(1-5)(4-5)(6-5)(2-5)(4-1)(6-1)(2-1)(6-4)(2-4)(2-6) is negative therefore the term is negative. On the other hand if the column element permutation is a11a22a33a44a55a66 then the product of index differences (2-1)(3-1)(4-1)(5-1)(6-1)(3-2)(4-2)(5-2)(6-2)(4-3)(5-3)(6-3)(5-4)(6-4)(6-5) is positive.

Antonio,reading through this opening thread of yours,I had visions of Ying and yang,
why do you think this was?

regards michael.

[AntonioLao]

I had visions of Ying and yang, why do you think this was?

Energywise it is always the positive that we can work with. The trick is to make the negative positive then we get limitless source.

[mkirkpatrick]

Energywise it is always the positive that we can work with. The trick is to make the negative positive then we get limitless source.

There is always the alternative of boring through,tunneling might do the job!





regards michael/