Jump to content

Why bother diagonalising matrices???


Recommended Posts

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?

Link to comment
Share on other sites

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

Link to comment
Share on other sites

  • 1 month later...

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

Link to comment
Share on other sites

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: ex

diagonalizing a matrix A makes it easy to

find eA

Link to comment
Share on other sites

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

Link to comment
Share on other sites

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 eA

 

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.

Link to comment
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
×
×
  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.