Why bother diagonalising matrices???

[bloodhound]

We all know how to diagonalise matrices. But ever since i leanrt how to I was like WTF do u use it for????

 

Until now, the only really useful application is to write any general quadratic form as a sum of perfect squares.

 

Any of you guys know any?

[stevem]

One example is to classify quadrics (3 or more dimensions) or conics (2 dimensions)

 

For example, suppose you want to know which type of quadric [math]5x^2+3y^2+3z^2-2xy+2yz-2xz-10x+6y-2x-9=0[/math] is.

 

In geometrical terms you rotate it and translate it so it has one of the standard forms listed in quadrics

 

In algebraic terms, you put the equation into the form [math]\mathbf{x}^T\mathbf{Ax}+\mathbf{J}^T+H=0[/math] where [math]\mathbf{A}[/math] is a [math]3 \times 3[/math] matrix, [math]\mathbf{J}[/math] and [math]\mathbf{x}[/math] are column vectors and [math]H[/math] is a real number.

 

You then diagonalize [math]\mathbf{A}[/math] to get [math]\mathbf{P}^T\mathbf{A}\mathbf{P}=\mathbf{D}[/math] where [math]\mathbf{P}[/math] is an orthogonal matrix. Then you transform the equation using [math]\mathbf{P}[/math] and that effectively gives you new perpendicular axes, which is in effect a rotation. After that completing the square gives you a translation and you end up with [math]\frac{x^2}{3}+\frac{y^2}{6}+\frac{z^2}{9} =1[/math] which is an ellipsoid

[matt grime]

try finding the n'th power of a matrix for n large, then, if possible diagonalize, repeat the operation and see why. also seeing as people prefer applied reasons, you can find the direction and magnitude of max min strain, resistance etc in tensors from the diagonaliztions and the eigenvectors/values

[Martin]

try finding the n'th power of a matrix for n large, then, if possible diagonalize, repeat the operation and see why. also seeing as people prefer applied reasons, you can find the direction and magnitude of max min strain, resistance etc in tensors from the diagonaliztions and the eigenvectors/values

 

expanding on what matt grime just said,

you know that so often with ordinary numbers x

one wants to know the exponential: e^x

diagonalizing a matrix A makes it easy to

find e^A

[bloodhound]

how is an exponential of a matrix defined

[Martin]

with a diagonal matrix you just take ordinary exponential down the diagonal

 

 

but more generally it is a power series

 

you know how exponential is ordinarily defined:

 

exp(x) = 1 + x/1 + x²/2! + x³/3! + ...

 

well that extends to n x n matrices just by interpreting the first term 1 as the n x n identity I

 

exp(A) = I + A/1 + A²/2! + A³/3! + ...

 

btw cartman was a great idea (and the whole bubbles thing)

I still like cartman best although zoidberg is also very fine

[JaKiri]

Re: the bubbles thing, I suspect (and indeed know) that it gets very old very fast.

[bloodhound]

thanks jakiri. i learned that after i googled it. thanks anyway. quite interesting.

[fourier jr]

expanding on what matt grime just said' date='

you know that so often with ordinary numbers x

one wants to know the exponential: e[sup']x[/sup]

diagonalizing a matrix A makes it easy to

find e^A

 

I thought that was the craziest thing when I learned about it in a DEs course. Then I saw how important & usefule it is to do that. Luckily there's a way to deal with that without doing an infinite sum of powers of matrices, but I can't remember what it is.