permutation matrices

[AntonioLao]

The permutations of N objects taken N at a time is simply N factorial: N!=N(N-1)(N-2)…(N-N+1). According to this rule, for three objects, N=3 and 3!=(3)(2)(1)=6. Three objects A, B, and C arrange in row matrix form becomes (ABC) where A is in row 1 column 1, B is in row 1 column 2, and C is in row 1 column 3. Using the same construction, the 3 by 3 identity matrix is written as I(123). The 1st number inside the parenthesis indicates that at column1 row 1, the matrix element has a value of 1, likewise in column 2 row 2 and column 3 row 3. The remaining off diagonal elements has values of zero. In the same context, the 3 by 3 matrix M(312) indicates that 1’s can be found at row 3 column 1, row 1 column 2, and row 2 column 3. The product of (ABC) by M(312) is (CAB). The other permutation matrices are: M(132), M(213), M(231), and M(321). When these are multiplied to the left by (ABC) then the final four respective permutations are: (ACB), (BAC), (BCA), and (CBA).

It can be stated without proofs that these six permutation matrices form a permutation group if and only if they satisfy the three group properties: (1) matrix multiplication is associative, (2) there is an identity matrix such that its products with the other matrices give the same matrices, and (3) for each matrix there exist an inverse matrix such that their product gives the identity matrix. Clearly, I(123) is the identity matrix and property 2 is satisfied. It can be shown that M(123), M(132), M(213), and M(321) are self-inverses, that is each multiplying by itself gives the identity matrix and property 3 is satisfied. Consequently, property 1 is also satisfied. Therefore these six permutation matrices: M(123), M(132), M(213), M(231), M(312), and M(321) form a permutation group.

[labelwench]

http://en.wikipedia.org/wiki/Permutation_matrix

Read it. Stared at it. Read it. Stared at It.

Walked away.

You seem to post a new thread every day, and have posted more threads than any other at the forum.

To what purpose?

[AntonioLao]

To what purpose?

to arrive at a quantum theory of the space-time continuum justifying the experimentally proven existence of quantum vacuum fluctuations as zero-point energies, which can help understand cold fusion and subsequent infinite energy extraction from the quantum vacuum.

[labelwench]

to arrive at a quantum theory of the space-time continuum justifying the experimentally proven existence of quantum vacuum fluctuations as zero-point energies, which can help understand cold fusion and subsequent infinite energy extraction from the quantum vacuum.

So, in effect, you are casting bread upon the waters, hoping that in time, the right person will come across one of these threads, and respond in a manner that will allow you to progress with your work in the direction of cold fusion. Now it makes more sense.

Do you post in other forums as well?

Have you encountered any who have been of assistance with this theme?

[AntonioLao]

respond in a manner that will allow you to progress with your work

You are absolutely correct but by posting it also help me solidify the different ideas. I used to post in the Physicsforum, since I don't have my own computer anymore and using the limited access at the public library I can only post 1 hour more or less each day when the library opens.

[labelwench]

You are absolutely correct but by posting it also help me solidify the different ideas. I used to post in the Physicsforum, since I don't have my own computer anymore and using the limited access at the public library I can only post 1 hour more or less each day when the library opens.

Then it is fortunate that your library has public access, as has ours, although the demand and the limited use can be frustrating.

These days, computers are becoming easy to acquire as many will give them away when they upgrade their equipment and a net-book can be purchased new for $300.00, which is not suitable for gamers, yet excellent for surfing the net and forum work. It is the service provider that gets costly over time and our access for high speed inter-net (high speed being relative, lol) is about $66.00/month, and it is only in recent years that a combined income would justify such expense. Until about 6 years ago, we used our library a lot, as they also loan movies and documentaries, as well as books, and the winters are long in these parts.

Yes, most of my life I have had to be very astute with funds, and given the present economic uncertainty, those lessons may well come in handy at some time in the future....

[AntonioLao]

If I can afford it I would rather have my own computer at home.

[labelwench]

If I can afford it I would rather have my own computer at home.

The time required for the earning of the medium of exchange (money) and then also having sufficient time and energy to enjoy the interactions which the application of the earned medium may afford are cause of consternation to many.

How many work and save through their lifetime, only to lose either their savings or their health, and never enjoy the fruits of their labor? Too often have I witnessed such.

With that in mind, I strive ever to balance my efforts with the outcomes, yet last night, for the first time ever, I clocked out 3 hours before my shift was done. My main tasks were accomplished, and I could not clear my mind of the notion that curling up anywhere (even the floor looked inviting!) and sleeping, was what this organism valued the most at that moment.....

Fortunate am I, that I can afford to make the decision that was in the best interests of my continued health. I drove home most cautiously, aware that I was in a dysfunctional state, and went to bed straightaway, arising only shortly....

Perhaps in future, all homes will be designed with built-in computer and inter-net access, the way most enjoy appliances these days?

[AntonioLao]

The truth is we cannot all be millionaires and although the Chinese believed in "luck," the simple secret of making millions or more is not known to everyone. Nonetheless, the progress of science and technology does allow people of the future to live comfortably but will they live peacefully is another question.

[labelwench]

The truth is we cannot all be millionaires and although the Chinese believed in "luck," the simple secret of making millions or more is not known to everyone. Nonetheless, the progress of science and technology does allow people of the future to live comfortably but will they live peacefully is another question.

There cannot be the expectation of peace when the opportunities for a few come at the expense of the many, and our lack of understanding that the resources of this planet are conjoined in an intricate balance. Our activities and intervention into these processes of which we know but little, may have consequences that we shall lack the ability to mitigate.

Change is a natural process and all organisms are faced with the challenge to adapt. Should the pace of change accelerate beyond the ability to adapt, there is but one outcome on historical record.