I’m one of those unfortunately literal sorts who, if he can’t see a practical use for something, has great trouble learning it. I recall such a time many years ago when I took a math class that turned out to be almost entirely about matrices. At the time I was too shy to ask the prof, “What’s the point of this? What will I use it for?” so the semester was the purest misery and I felt fortunate to come out of it with a C-, maybe half a point above a D, so I didn’t have to take it over to graduate. And, much like sine waves, as I had no idea at the time I might eventually need to know how to use them I would probably have dismissed their applications as irrelevant to my life.
So, all this time later I turned out to be a sorta audio engineer so it’s back to using (and relearning) wave functions and I want to head off any potential matrix problems BEFORE I’m expected to be able to use them. From what quarter can I expect to be surprised?
Your instructor must have done a poor job if he/she didn’t convey why the subject is important. I’ve always viewed matrix calculations as the computational part of linear algebra. Probably the most useful application of matrices is in solving or reducing systems of linear equations, which can also lead to solving optimization problems. I really don’t think that you’ll be surprised by any "matrix problems’ because most solutions that require matrix calculations have already been computerized. For example, you may have to find a trend line or a multiple regression line at some point. But you really don’t have to know how to do the matric calculations to get the answer – all that number crunching will be done for you in the software that is readily available.
Matrix calculations are used in pretty much any field you can think of. In particular, multivariable statistics relies heavily on matrices, so any field that uses multivariable statistics will use matrices.
Any chance of elaborating with an example nivlac?
What real problem would require matrices (if we didn’t have software that could do it for us) and how would they allow us to arrive at a solution?
BTW, I had a fantastic maths teacher, but he had no enthusiasm for matirces at all; I’m sure he wouldn’t have bothered teaching them if they hadn’t been part of the core curriculum.
Matrices are used for computer graphics. There are many different coordinate systems involved, and matrices allow the computer to quickly determine which color goes in each pixel on the screen.
A vector is just a group of numbers, so if something can be described by several numbers, you can just use one vector. And any calculation using this vector is done with a matrix.
One example: the polarization state of light requires 4 numbers to describe in full (“Stokes parameters”), so you write it out as a 4-element vector. An optical element (polarizer, lens, mirror, etc) modifies the polarization state of light, and if it is linear (double the input and the output doubles), you can use a matrix to describe that element. So it’s a simple matter to define polarizatio of an input beam, multiply it with all the matrices describing all the elements in the instrument, and the resulting vector describes the polarization of the output beam.
That’s just one example, one I happen to use every day. I’m sure there are many other instances where the status of something is described as a vector, and any operation/modification is described as a matrix.
Well, if you had a system of linear equations of different variables, you could use matrix methods to solve them:
a1x + b1y + c1Z = A
a2x + b2y + c2z = B
a3x +b3y + c3z = C
If you know a1, a2, a3, b1, b2, b3, c1, c2, c3, and A, B, and C, you can use 3 X 3 matrices to solve for x, y, and z. You can do it without matrices, too – it’s a lot of algebra. But it’s more orderly with matrix form,alism (and ultimately, the same procdure, really). Matrices also give you simple ways to rotate objects about three axes – computer software that rotates objects uses rotation matrices somewhere in its guts.
Matrices pretty naturally show up in the physics of mechanics and materials science in relating the directional properties of materials. You can perform paraxial optics with 2-dimensional matrices. They show up in quantum mechanics in many applications. But we’re getting esoteric – you can visualize the need for matrices in solving rotational problems, or in systems of linear coupled equations.
One way to model a electical circuit is with a matrix.
Its been a LONG time but for the (i,j) entries is the admitance between nodes i and j and (i,i) is the sum of branch tree admitances to ground (or whatever the reference node is).
How do you think the software does it? The nice thing about matrix calculations is that they lend themselves very well to implementation in computer systems.
Yes, this may be the first classic use for matrices that most people learn: to solve systems of linear equations. (Actually, there’s more than one method that uses matrices: Gauss-Jordan elimination, using inverse matrices, Cramer’s rule…) Systems of differential equations are also treated using matrices.
Right. Matrices define linear transformations, such as rotations and reflections, that get used in computer graphics and elsewhere.
Matrices are also used in linear programming problems and in working with the Leontief input-output model in economics, to name just a couple more examples.
I don’t, and will never, need to know how to use matrices because if people start talking about stuff like you are talking about I am extremely unlikely to be in the room.
When I do happen to use matrices, like in CAD, I have software to think about it for me.
For my own, personal sense of satisfaction I not only have to go back to school to study math but also need to tell the college people to ignore the credits I transferred and let me take classes I supposedly took already because I really, REALLY didn’t understand what was being taught.
However, it would be a cheaper and less frustrating use of my time if I were to accept that some things will forever remain beyond my ken and that I should give up my last hopes of ever being smart or even knowing how to use more than half the buttons on my calculators (which is especially pathetic since I collect the damned things) much less how to do stuff that you can’t do on a calculator.
Well, I think what we’re saying is that matrices are super useful in many fields. There is no harm in understanding how they work, how to do the calcs, and some simple applications. But, in practice, real life applications will invariably have all the computational aspects of matrix math programmed for you. I think if you want to pursue a deeper knowledge of matrices, the best way is to focus on a specific field, such as computer animation, statistical analysis, differential equations, linear programming. I should add that matrix math can also provide a very compact, elegant notation for expressing complex relationships between variables.
The use of matrices includes, but is not limited to situations described in everyone’s posts here. I thought by now you’d have some idea of how versatile and generic a tool it is.
That’s a little like saying “I don’t need to understand what addition and multiplication are, because my calculator does it for me.” Well yes, your calculator can do the actual question, but you still need to understand the concept so you can input the question correctly into the calculator and interpret the result correctly.
Matrices are used HEAVILY in digital signal processing. So, matrices were used a lot in designing and implementing any filtering or other processing used on digital media: DVD, CD, etc.
Others have given examples of the usefulness of matrices. As to hand solution of a matrix equation there are a couple of methods both of which are useful for up to a 3 X 3 matrix. Above that it gets pretty hairy. A matrix equation would look like this:
[M] = [Y] Where [M] is a square matrix, 2 X 2, 3 X 3, etc. and [Y] are both column matrices with one column and as many rows as there are rows in [M].
The most direct way to solve the problem is to construct a matrix, say [A] which conists of the elements of [M] with the first column replaced by the elements of [Y] if you are solving for x[sub]1[/sub]. Then compute the determinants of [M] and of [A] and divide the first into the second for the value of x[sub]1[/sub]
For x[sub]2[/sub] replace the second column of [M] with the elements of [Y] and repeat the above process, and so on for x[sub]3[/sub].
Alterntively you can compute the inverse of [M]. The inverse of [M] is a matrix such that
[M][sup]-1[/sup][M] = where is the unit matrix. That is a matrix in which all of the main diagonal elements are 1 and all the other elements are 0.
Multiply both sides of the original equation by [M][sup]-1[/sup] to give
[M][sup]-1[/sup][M] = [M][sup]-1[/sup][Y]
Perform the multiplications and you have the values of the x’s.
None of the above is all that easy for even a 3 X 3 but it can be done.
Students who are studying matrix methods should probably do each method a time or two so as to appreciated how wonderful the computer solution is.
When you use matrices for electrical networks it turns out that if you always use the same sign converntion for current and voltage polarities, you can write the impedance or admittance matrix by inspection. Saves an awfull lot of time.
Hah! You have no idea of the people who are usually in the same room as me. They think I’m a mathematical genius :eek: and are totally unlikely to bring up the topic of solving matrix problems.
I have a calculator that will do matrix calcs, though that sounds like the easy part. Now I just need (want? vaguely desire?) “to understand the concept so (I) can input the question correctly into the calculator and interpret the result correctly.”
Now that you guys have told me what they are good for, what are the specifics of how they are used? Let’s go with mirroring an object in CAD, since it’s something I tell my computer to do several times a day. And please treat me as somebody who barely squeaked through the class thirty years ago and forgot what little he learned ten minutes after the final; for all intents a total neophyte who has little or no idea what you are talking about. What David wrote, for instance, shot WAY over my head.
Matrices are used everywhere. As has already been mentioned, they’re used extensively in computer graphics, for example, there’s such a thing as a rotation matrix, which, when multiplied with a vector, will produce a new rotated vector (there’s other methods for rotation, aswell).
Matrices are also used a lot in coding theory, for example, self correcting codes rely heavily on matrix calculations (or can be formulated so ).
Easy. Suppose, for instance, we are working in one of the two dimensional plane views in a CAD program, and so every point can be represented as a pair of numbers, (x, y). Now, let’s suppose that we want to mirror in the y axis, so after multiplying our mirroring matrix with the original vector, we get (-x, y). Here is such a matrix:
-1 0
0 1
So, to mirror a point, we simply use the following formula:
new point co-ordinates = old point co-ordinates * mirroring matrix.
Of course, if we wished to mirror in another axis, the matrix would be different.
I still don’t see how the matrix is being used (or what the “original vector” is, for that matter), though I sometimes cheat in Autocad and mirror objects on the Y axis by changing the X scale to -1 so I get that part of it. I’ll stop bothering you guys and find someplace I can start at square one.
