I need help with maths :(

I'm sure theres plenty of people here who have done this sort of stuff before... this question has me stumped like no other before it.

Find a matrix P that orthogonally diagonalises
a b 0
b a 0
0 0 1

The fact theres variables in there makes it hard... Any help at all would be appreciated.

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"If you talk at all during this lesson, you have detention. Do you understand?"

• My yr11 Economics teacher

wow

ive done matrices but nothing that complicated

sorry

few years ago :smile: could it be:

1/(a?-b?)*

a.....-b.....0

-b....a......0

0......0...a?-b?

there's nothing complicated about variables, just take them with you in your calculations

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These words are my diaries screaming out loud

I know a fair bit about matrix algebra, but orthogonally diagonalises? I'll have to ask my teacher about that. Must be like hyperbolic sine or one of those other weird things that has no real purpose.

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Better to be in denial than to be human.

Bill Gates understands binary: his company is number one, and his customers are all zeros.

Linear Algebra ... been awhile ... grabs book

Well, If that matrix is matrix A and the diagonalize matrix is P, then:

P^-1 AP = P^T AP is diagonal. That's just a way to check.

So, these are the steps:

1) Find a basis for each eigenspace of A

2) Apply the Gram-Schmidt process to each of these bases to obtain an orthonormal basis for each eigenspace.

3) Form the matrix P whose columns are the basis vectors constructed in step 2; this maitrx orthogonally diagonalizes A.

At to the solution for your problem ... I'm not going to calculate it out right now ... it's pretty heafty, unless there's a trick I'm not seeing (could be an identity).

The answer to the det(lambda * I - A) problem should look something like: (lambda - 1)((lambda - a)² - b). You get the eiganvalues from that (don't use my solution, unless you calculate it and find it to be correct), use those values to find basis for the eiganspaces.

Then apply Gram-Schmidt process to {u1, u2} to get the orthonotmal eigenvectors. v1, v2, and vn are the columns for matrix P.

Look through your book and it should tell you exactly how to calculate it all, the rest of it relies on your ability to simplify :smile:

BlisTer, that's incorrect, since: P^T AP isn't diagonal.

See:

| a(a²-b²) -b(a²-b²) 0 |
| -b(a²-b²) a(a²-b²) 0 |
| 0..............0.(a²-b²)²)|

Anyway. I put up how to solve it ... not sure if your book covers it well though.

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Blame it on Microsoft, God does.

idd.

i thought it was a mindteaser and that we were just looking for A^(-1)
since, when multiplied with A that creates a diagonal, orthogonal
matrix (the unity matrix).

all starts coming back to me, but i'm not gonna look it up, i think Crono states it correctly.

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These words are my diaries screaming out loud

Crono is right... the problem is, it shouldnt be hefty. The only way I know is with eigenspaces, but to do that with these variables would be messy. Theres a trick to this somewhere, I'm sure.

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"If you talk at all during this lesson, you have detention. Do you understand?"

• My yr11 Economics teacher

<pedantic>Its det(A-lambda*I), not (lambda*I-A) since you are looking for solutions v to:
A*v = lambda*v, so:
A*v = lambda*I*v, subtracting:
A*v - lambda*I*v = 0, collecting:
(A - lambda*I)v = 0, assuming v is not the zero vector:
det(A - lambda*I) = 0
</pedantic>

Oh also hyperbolic sine is awesome.

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http://maze5.net

And thought my maths was complicated :biggrin: We just started on eigenvalues
and eigenvectors, but eigenspaces are entirely alien to me :smile: God I
hate maths...

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[img]http://card.mygamercard.net/sig/Default/reno84.png[/img]
Designer @ Haiku Interactive | ReNo-vation.net

And thought my maths was complicated :biggrin: We just started on eigenvalues
and eigenvectors, but eigenspaces are entirely alien to me :smile: God I
hate maths...

Yeah i must say that i thought the same.

Ive just started with complex numbers, and its quite f**ked up =P.

When you find an eigenvector, its not just one vector, there are a bunch of solutions to the equation. An eigenspace is the space containing all eigenvectors for that eigenvalue. Not as scary as it sounds :smile: I'm in first year, I cant see why they wouldnt teach you this stuff until 4th, particularly in a field like computer games. Yeah... and complex numbers are pretty weird, but not really all that difficult. Relatively speaking.

I cant find anyone else who can do this question... I have less than 2 days until the assignments due, if I find out how to do it I'll post it here :biggrin:

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"If you talk at all during this lesson, you have detention. Do you understand?"

• My yr11 Economics teacher

Mostly because my course sucks balls :biggrin: I'd also say that my maths
lecturers are our most competent, though one is a little too humourless
and patronising to the less capable in our class. Oh sad, sad truth.
Going by my mates in other universities around the UK, degrees in
computer science these days seem to boil down to courses in how to
program in java. Oh I'm not bitter and cynical...

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[img]http://card.mygamercard.net/sig/Default/reno84.png[/img]
Designer @ Haiku Interactive | ReNo-vation.net

God I
hate maths...

Ditto. I've never actually found any use for more than 90% of the complicated math we learn(ed) in school.

And they all yelled at me that I need math for making games and for flying. WELL GUESS AGAIN! :evilgrin:

I did A level maths and what you guys are talking about seems like jibberish to me!

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-[Better to be Honest than Kind]-

Yeah, Linear Algebra is a 300 level course here. And Orthogonal stuff isn't covered until Chapter 7 ...

As for saying it isn't used: HA HA HA. Yes, yes it is. You'll use it in graphics programming ... a lot. Not orthogonal crap, maybe, but, linear algebra as a whole: yes. That's the only reason they'd be teaching it to you now ... as for your problem ... again ...

lambda = a and 1 ... a, since it's lambda is squared, has two eiganvectors from one eiganspace ... and 1 has one eiganvector ... making up your answer. This was the only thing I had trouble with in my class actually, since it isn't really ... um ... explained how to get from eiganvalues to eiganvectors. I could look up some stuff.

You have to start with the eiganvalues ... a and 1
Use an arbritrary vector, x:

| x1 |
| x2 |
| x3 |

Take the matrix lambda * I - A found earlier and input lambda = a ... the matrix equals 0 as in

| a-a  -b    0  | | x1 |   | 0 |
| -b   a-a   0  | | x2 | = | 0 |
|  0    0   a-1 | | x3 |   | 0 |

This yeilds a relation.

-bx2 = 0
-bx1 = 0
a-1x3 = 0

You find the x's, they are related, find them as arbitrary values (s, t, -s ... whatever)

Then, you get the vector, by seperating and simplifying (this is just an example)

Let's say you got values, -s, t, and s, SO

| -s |   | -s |   | 0 |     | -1 |     | 0 |
|  t | = |  0 | + | t | = s |  0 | + t | 1 |
|  s |   |  s |   | 0 |     |  1 |     | 0 |

And there's two vectors ... get it? You do that for the eiganvalues, find all possibilities (there will be three total) and that's the orthogonal matrix P ...

I hope that font works and lines everything up ... otherwise that might be confusing.

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Blame it on Microsoft, God does.

Damn! All I know about higher maths is the 3D space... damn... I can
code opengl and all but matrixes... are new to me :biggrin: I never needed
them... I mean like... it is real easy to place vectors in 3D space you
know... but can anyone explain me the practical use of matrixeS?

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HL2 tutorials 'n' stuff: http://madedog.pri.ee
217.159.236.34:27050 - CSS Server - Clean | koffer.ee

if I find out how to do it I'll post it here :biggrin:

Crono has explained to you how to do this... twice. Other than the fact that he's solving for the eigenvalues in a bizarre way that I'm not entirely convinced works, the procedure he describes is exactly correct for orthagonally diagonalizing a matrix.

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http://maze5.net

It's how my book goes about it, actually. It seems to work, there's just a bit of guess work, but it's doable. The procedure works ... I probably didn't explain it well enough though ...

Madedog, matricies and matrix math comes in when conceptualizing ... how do you think OpenGL was written? It does the work for you.

It's kind of like dot product, I was like, "Wow, I'm glad I'll never have to use that again" ... wrong. It's heavily used in shaders.

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Blame it on Microsoft, God does.

Thanks Crono, your second post there cleared things up. I'll try out what you did. I was a bit reluctant to actually work out the eigenvalues because it looked like the variables would make it messy :/

Maths is the most useful thing you can learn.

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"If you talk at all during this lesson, you have detention. Do you understand?"

• My yr11 Economics teacher

Not a problem, it actually took awhile for me to find a proper example in my book that showed how to get from eiganvalues to eiganspaces (lambda * x)/vectors. But, I remember this a little better now. It just seems you've got a funky matrix to deal with. If it had some better values it'd be a five minute type deal. It's doable though.

Anyway, doesn't your book cover this? Seriously. I'd be really surprised if your instructor would give you a problem like this if it wasn't explained well.

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Blame it on Microsoft, God does.

So heres the standard (in my experience) way to solve for eigenvalues and eigenvectors:

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The meaning of eigenvectors and eigenvalues:

The definition of an eigenvalue eigenvector pair for a given matrix is that when you hit the eigenvector with the matrix, you get the same thing when you multiply the vector by the eigenvalue.
This is written like so:
matrix*eigenvector = eigenvalue*eigenvector, or
A*v_eig = lambda*v_eig.

Basically when you take any vector 'v_initial' and "hit it" with a matrix 'A',
ie: A*v_initial = v_final,
the vector 'v_initial' is transformed into a new vector 'v_final'. Depending on the matrix, this transformation can be a reflection about a line, a rotation, a scaling, combinations of those things, etc...

Geometrically eigenvectors are just a special case where the vector is only scaled - no rotations, reflections, etc. You hit v_eig with A, and you get out v_eig multiplied by a scaling factor.

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Ok, so now onto actually solving for eigenvalues and eigenvectors (witout the guesswork):

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Finding Eigenvalues:

We start with the basic definition for eigenvectors and eigenvalues:
A*v_eig = lambda*v_eig

We want to somehow solve for lambda from that. So first we might try subtracting lambda*v_eig from both sides of the equation.
A*v_eig - lambda*v_eig = 0_vector

Here we get mildly excited since we notice that both of the terms have a v_eig in them. It would be super-nice if we could factor out that v_eig from both terms. Unfortunately we can't factor it as (A-lambda)*v_eig, since you can't subtract a scalar (lambda) from a matrix (A).

The trick is to remember that if you hit any vector with the Identity matrix, you get back the origiona vector. Using this fact, we can write the equation as:
A*v_eig - lambda*(I*v_eig) = 0_vector, or
A*v_eig - (lambda*I)*v_eig = 0_vector

This we can factor, and we get
(A - lambda*I)*v_eig = 0_vector

Now we have a matrix hitting a vector, and we get out the zero vector. If you recall from matrix multiplication and determinant properties, this can happen in 2 cases:
1: v_eig is the zero vector (not very interesting), or
2: the determinant of (A - lambda*I) is zero.

The 2nd case is interesting because it gives us a nice equation that we can use to solve for lambda:
det(A-lambda*I) = 0

All you have to do is calculate that determinant, set it equal to zero, and solve for lambda. You will likely notice that the equation you get will have multiple "roots" ie: you will be able to write it as
(lambda - x0)(lambda - x1) (...) = 0.

These multiple solutions (lambda_1 = x1, lambda_2 = x2, ..., lambda_n = xn) are all valid and each one will correspond to an eigenvector.

Ok, now on to finding those eigenvectors that they correspond to:

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Finding eigenvectors given their eigenvalues.

Once you have the eigenvalues, you can just plug them into the equation:
(A-lambda*I)*v_eig = 0

But thats just a system of linear equations you can solve by row-reduction. So all you have to do is do row-reduction on the matrix (A-lambda*I), and you will get out the components of the eigenvector.

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http://maze5.net

Maze ... that's exactly the method I described. Pay more attention :razz:

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Blame it on Microsoft, God does.

You were doing lambda*I - A, not A - lambda*I

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http://maze5.net

That's the only difference though, saying the guess and check stuff didn't make sense then.

Can someone explain why it's that way in my book and in the class I had ... and other books, then? I'm sure both give the answer you need. Actually ... yeah, they are both right. When I did it you end up with negative numbers throughout the matrix, wonder which is preferred ... since they're both right. However ... if you do it the way I did it there'd be no re-arranging once you get ready to factor ... if you need to rearrange for personal reasons anyway.

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Blame it on Microsoft, God does.

OK, I've done it. Heres how I did it (pretty similar to whats been posted here already):

Find eigenvalues, solve for det(lambda*I - A) = 0 (if you do A-lambda*I, it will work out the same, detA = det(-A)). This got a bit tricky, I used the quadratic formula to solve. Then eigenvectors, I did almost exactly what crono did. G-S from that to get ON bases, and put them into a matrix.

The answer is: P=
0 1/sqrt(2) 1/sqrt(2)
0 -1/sqrt(2) 1/sqrt(2)
1 0 0

Thanks for the help guys. I was thinking there must be a trick to this... apparently not.

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"If you talk at all during this lesson, you have detention. Do you understand?"

• My yr11 Economics teacher

So, uhm... Why's there letters in your math problem?

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I tried sniffing coke, but the ice cubes kept getting stuck in my nose.
http://www.dimebowl.com

Because letters are harder to deal with than numbers. Since when you deal with numbers they usually cancel ... a lot.

Wil5on! I forgot to mention to use the quadratic formula, sorry. That's right, because using lambda = a only gave you one value, right? Which is the 001 vector. The quadratic formula would give the other two ... yeah ... I think that's it.

Anyway, nice. You solved it: yay. Now, cross it back to make sure it's correct.

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Blame it on Microsoft, God does.

Jesus this is serious maths! I'm poor with maths and the best i'm learning is Algebra and nth term hehe!

ha! we just finished triginometry! it was pretty easy actually, just a lot of cosine, tangent and sine ratios. i would have to say maths is one of my strong points but 'linear algebra'? im a little curious maybe ill look into a little more.

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Bullet Control: $5000 for a bullet.
"I would blow your f**king head off! ...if I could afford it. I'm gonna get another job, start saving some money... then you a dead man!"

It comes after Calculus (Like geometric series stuff). Or ... depending on how far you go in Calculus ... in the middle. It's up there with group and number theory ... bleh.

Actually, if I recall, chapter three in my book is Calculus 4 at my school ... it all starts blending and bleeding together once you get high enough.

Oh and if you're wondering, trigonometry is pre-calculus. Actually, quite literally, I haven't seen a class called "trigonometry" in a long time.

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Blame it on Microsoft, God does.

lol

im still in college al and anal math

of course im only a junior in highschool

Dude, there are seniors in my class, which happens to be like a
freshmen course (im a freshman, stop laughin, i know you guys are)

^_^, I just got onto matrices today, maybe I could help with a quick look through my book.

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http://img362.imageshack.us/img362/8521/windows981dk.jpg

Nickelplate is my dad

Here we do linear algebra and calculus simultaneously. Usually calculus is the hard part... differential equations, limits and stuff. Trig, at this level, is part of calculus.

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"If you talk at all during this lesson, you have detention. Do you understand?"

• My yr11 Economics teacher

wil5on said:
Here we do linear algebra and calculus simultaneously. Usually calculus is the hard part... differential equations, limits and stuff. Trig, at this level, is part of calculus.

we got all the algebra in the first year along with general calculus, statistics and numerical maths. differential equations in the second year. Those were all complemented with the applied courses for engineering. Numerical modelling was in my 4th year.

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These words are my diaries screaming out loud

Crazy. Linear Algebra here is like Post-Calculus 3 ... or something as such. However, It makes sense to learn Lin. Algebra last ... since with it differential equations become nonsense. For the most part anyway.

I Guess they only teach you what you need to know then. Not here ... it's like, "Hmm ... go learn everything then come back" it really sucks, because not only do you not remember it all, but it becomes harder to conceptualize. Until you get farther along that is.

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Blame it on Microsoft, God does.

Dude, there are seniors in my class, which happens to be like a
freshmen course (im a freshman, stop laughin, i know you guys are)

^_^, I just got onto matrices today, maybe I could help with a quick look through my book.

yeah its been like that for me since then too

what level of math are you in