bloodhound Posted June 4, 2004 Share Posted June 4, 2004 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 More sharing options...

stevem Posted June 5, 2004 Share Posted June 5, 2004 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 More sharing options...

matt grime Posted July 9, 2004 Share Posted July 9, 2004 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 More sharing options...

Martin Posted July 9, 2004 Share Posted July 9, 2004 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} Link to comment Share on other sites More sharing options...

bloodhound Posted July 9, 2004 Author Share Posted July 9, 2004 how is an exponential of a matrix defined Link to comment Share on other sites More sharing options...

Martin Posted July 9, 2004 Share Posted July 9, 2004 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 + x^{2}/2! + x^{3}/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 + A^{2}/2! + A^{3}/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 More sharing options...

JaKiri Posted July 9, 2004 Share Posted July 9, 2004 Re: the bubbles thing, I suspect (and indeed know) that it gets very old very fast. Link to comment Share on other sites More sharing options...

bloodhound Posted July 9, 2004 Author Share Posted July 9, 2004 thanks jakiri. i learned that after i googled it. thanks anyway. quite interesting. Link to comment Share on other sites More sharing options...

fourier jr Posted July 10, 2004 Share Posted July 10, 2004 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. Link to comment Share on other sites More sharing options...

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