Matrices and Maths

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Okay I haven't taken/used Linear Algebra for roughly four+ years now.

But I was thinking about it the other day and remember that matrices are used to represent linear equations and I remember cross products, dot products, and determinants.

But I only vaguely remember the PURPOSE of those things and I only don't recall if matrices let you do things you couldn't do with regular algebra for linear equations, or only helped you to solve them more efficiently.

I ask also becuz I remember reading a long time ago that certain fancy quantum things could only be accomplished with matrices and not regular maths and that limited the amount of people who could understand them. Is this true? (I unf. never took Quantum in school coz I was somewhat rubbish at physics and the first two courses which went thru electromagnetism etc were plenty rigorous for me)

― Sterling Clover (s_clover), Friday, 7 March 2003 18:25 (twenty-three years ago)

I have no idea about the quantum stuff, but matrices make solving large systems of equations much, much easier (from a computer programing standpoint)

― fletrejet, Friday, 7 March 2003 18:38 (twenty-three years ago)

Matrices are kinda necessary if you want to pursue any career dealing with implementing 3D technology.

― donut bitch (donut), Friday, 7 March 2003 18:39 (twenty-three years ago)

more specific:

what's the point of a determinant?

taking a cross product is usually to substitute in a set of values to an equation system, right?

but then when/why do you take a dot product?

― Sterling Clover (s_clover), Friday, 7 March 2003 18:41 (twenty-three years ago)

Well, one of the many uses of a determinant is to use Kramer's rule to solve a system of equations.

― fletrejet, Friday, 7 March 2003 18:48 (twenty-three years ago)

i mean as i recall we learned how to get a determinant and how to use it a few different ways, but never got a grasp on WHAT or WHY it should be such a thing.

― Sterling Clover (s_clover), Friday, 7 March 2003 18:53 (twenty-three years ago)

blimey i used to know all this stuff

― mark s (mark s), Friday, 7 March 2003 19:09 (twenty-three years ago)

Me too.

Depressing isn't it?

No, wait a minute, it's quite refreshing that I don't know it any more.

― mei (mei), Friday, 7 March 2003 19:31 (twenty-three years ago)

No, wait a minute, it's quite refreshing that I don't know it any more.

This just makes me depressed that I have to battle with this stuff. Hotelling deflation anyone?

― jellybean (jellybean), Friday, 7 March 2003 19:53 (twenty-three years ago)

matrices are used to represent linear equations

They are used for systems often represeting linear equations in their rows.

― Mr Noodles (Mr Noodles), Friday, 7 March 2003 20:48 (twenty-three years ago)

They are used in modern physics to represent tensors, sorta multidemensional matricies.

Tensors are NOT fun.

― Mr Noodles (Mr Noodles), Friday, 7 March 2003 20:51 (twenty-three years ago)

There also great for keeping track of eigenvalues and eigenstates and while its easier to use linear algebra then calculus sometimes, its not a requirement. Its been two years so Im pretty rusty on the topic and was never that good at in the first place.
Its funny you brought that up as I gotta remeber to take my Quatum Mechanics book home.

― Mr Noodles (Mr Noodles), Friday, 7 March 2003 20:54 (twenty-three years ago)

if you take the determinant of a matrix you can find out if it's invertible or not, which tells you tons of stuff about it

also, dot products are useful for finding out if things are linearly independent or dependent

― liz p.., Friday, 7 March 2003 23:35 (twenty-three years ago)

matrices are very usful for fields (matlab is all based on matrices)

In maxwell's equations:
the cross product is used in faradays law and Ampere's law, and the dot product is used in Gauss's and the law of conservation of Charge
the curl of A is represented as:
del x A = det[(ax ay az) (d/dx d/dy d/dz) (Ax Ay Az)]

Faraday's law: del x E = -dB/dt
where E is the electric field and B is magnetic flux density

Ampere's law: del x H = J + dD/dt
where H is the magnetic field and D is the Displacement flux density

Gauss's law: del . D = rho
where rho is the volume charge density

Law of Conservation of Charge: del . B = 0

These four equations known collectively as Maxwell's equations are pretty much the backbone of all electro and magnetic fields behavior.

― A Nairn (moretap), Friday, 7 March 2003 23:57 (twenty-three years ago)

and divergence of A is: del . A
which is in Cartesian is dAx/dx + dAy/dy + dAz/dz

― A Nairn (moretap), Saturday, 8 March 2003 00:04 (twenty-three years ago)

in vector algebra
A . B = |A| |B| cos a
A and B are two vectors and |A| and |B| are their magnitudes and a is the angle between them.

similarly
A x B = |A| |B| sin a N
where N is the unit vetor in the direction depending on the right-hand-rule (if you point you fingers along A and curve to B your thumb points in the N direction

― A Nairn (moretap), Saturday, 8 March 2003 00:09 (twenty-three years ago)

liz is OTM. find the determinant for inversion. inversion of a matrix is really useful for solving the matrix with different inputs without having to do lots of recalculations. i used to know a lot more about this stuff, but it's leaked out of my head over the years.

― cprek (cprek), Saturday, 8 March 2003 01:21 (twenty-three years ago)

so if you take the matrix and find a P where P is invertible and say
M = PAP^(-1)
where A is a diagonal matrix, it makes it way easier to figure out what M to some big power is

― liz.. p, Saturday, 8 March 2003 10:03 (twenty-three years ago)

i think the geometric ways of defining this stuff are the nicest. so, e.g., the determinant of a matrix is the volume of the parallelotope generated by its columns, with a sign thrown for its orientation. and the dotproduct lets you project from one vector to another. This lets you understand some things rather easily, for instance why you'd want the determinant to be nonzero in order to solve a system of linear equations. But other stuff, such as i dunno cramer's rule, is still obscure from this point of view.

― dave k, Saturday, 8 March 2003 14:10 (twenty-three years ago)

Also it causes blood to spurt from your ears once there are more than three columns.

Matrices are one of the first ways in to True Science: Find diffucult problem, develop a bunch of tools that don't change the thing you want to solve, use these tools to change it to a simpler problem. It's all about invariants.

― Andrew Farrell (afarrell), Saturday, 8 March 2003 14:32 (twenty-three years ago)

I don't really remember using matrices for maxwell's equations -- more like solving rilly simple three dimensional integrations usually over the volume of a sphere, then twisting my hand around a bit to apply the right hand rule.

― Sterling Clover (s_clover), Saturday, 8 March 2003 15:13 (twenty-three years ago)

You could use them but quite often there is no need to go through it for something simple like Maxwell's equations. Its just a different way of representing vectors.

Matricist = new rockist of the mathworld.

― Mr Noodles (Mr Noodles), Saturday, 8 March 2003 18:28 (twenty-three years ago)

Hm, I wonder if it will catch on here the way 'rockist' has?

― Martin Skidmore (Martin Skidmore), Saturday, 8 March 2003 18:33 (twenty-three years ago)

It has a much more limited scope, I'll have to try it out at the next physics conference I attend.

― Mr Noodles (Mr Noodles), Saturday, 8 March 2003 20:43 (twenty-three years ago)

There really are a hell of a lot of mathematicians round here.

Anyone know any group theory?

― mei (mei), Saturday, 8 March 2003 21:31 (twenty-three years ago)

Is that about sex?

― Martin Skidmore (Martin Skidmore), Saturday, 8 March 2003 21:32 (twenty-three years ago)

Not enough to do your homework.

― Mr Noodles (Mr Noodles), Saturday, 8 March 2003 23:01 (twenty-three years ago)

or anyones for that matter, not accusing you of anything.

― Mr Noodles (Mr Noodles), Saturday, 8 March 2003 23:02 (twenty-three years ago)

Matrices are the main reason most RISC processors have a fused multiply-add instruction.

― Dave Fischer, Saturday, 8 March 2003 23:28 (twenty-three years ago)

You sure thats RISC, the whole idea was to get rid of those type of statements. Vector processors were created to handle simutaneous calculations on seperate data though.

― Mr Noodles (Mr Noodles), Saturday, 8 March 2003 23:52 (twenty-three years ago)

Ultra-minimalism may have been the original idea, back when RISC-I (commercialized as SPARC) and MIPS were academic projects, but it quickly changed to "load/store and fixed length instructions", which does rules out the excessively CISC stuff of VAX, 68K and IBM360 like string searches and linked-list operations, but allows for mul-add, if you have enough registers (which they do). MIPS, POWER, PowerPC, PA-RISC, some SPARC64 have a floating point multiply-add instruction.

The whole "RISC = load/store & fixed-length instructions" argument is why some people say the CDC-6600 (introduced in 1964) was the first RISC processor.

― Dave Fischer, Sunday, 9 March 2003 00:21 (twenty-three years ago)

I just don't think because a chip is RISC means it has a single instruction stream and multiple datastreams. Array processors like ILIAC IV (late 60s) havent been followed to well.
Vector processors though like CRAYs have been HUGE. I mean, who doesn't want a CRAY, powerbill not withstanding.

― Mr Noodles (Mr Noodles), Sunday, 9 March 2003 00:39 (twenty-three years ago)

Though I think we are talking about different things.

― Mr Noodles (Mr Noodles), Sunday, 9 March 2003 00:45 (twenty-three years ago)

I tend to think of SIMD machines as being a genre entirely unto themselves, and not related to the CISC-v-RISC debate. Many interesting features (RISC, SMP, pipelining, superscalar) can be found in machines from the 50s and 60s that were "rediscovered" in the 80s by microprocessor designers.

The fused mul-add instructions I'm talking about are just a single a + b * c, not an array operation.

The main thing that makes the Cray vector machines scream is actually the memory system. A fully configured T90 (mid 90s) did something like 900 gigabytes/sec to main memory.

― Dave Fischer, Sunday, 9 March 2003 00:58 (twenty-three years ago)

In that case we are both right because we are indeed thinking about different things.

Im not getting into a discussion on CRAYs today cause the leaf game is on. The memory was a big part of it and there were several changes compared to contempary designs including the wonderful multiple data streams which helps increase the memory access time.

― Mr Noodles (Mr Noodles), Sunday, 9 March 2003 01:04 (twenty-three years ago)

Is that about sex?
-- Martin Skidmore (martin****************), March 8th, 2003.

Ha!

If it was I'd be the Luuuuuurve Doctor.

But it's not. Damn.

― mei (mei), Sunday, 9 March 2003 10:37 (twenty-three years ago)

Anyone know any group theory?
A bit. Or at least I used to.

― OleM (OleM), Sunday, 9 March 2003 16:47 (twenty-three years ago)

Essential to any understanding of vibro-acoustics in multi-degree of freedom systems.

― Ed (dali), Sunday, 9 March 2003 16:50 (twenty-three years ago)

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