Mellinia Posted November 2, 2011 Share Posted November 2, 2011 Does the matrix[latex]\left[ \begin{array}{cc} x&y \end{array} \right][/latex][latex]\left[ \begin{array}{ccc} a&b&c\\d&e&f\\k&m&n \end{array} \right][/latex][latex]\left[ \begin{array}{c} x\\y\\1 \end{array} \right][/latex] has a name? I believe it opens to become [latex]{a{x^2}+e{y^2}+(b+d){xy}+(k+c){x}+(f+m){y}+n}[/latex] Link to comment Share on other sites More sharing options...
the tree Posted November 2, 2011 Share Posted November 2, 2011 (edited) Does the matrix[latex]\left[ \begin{array}{cc} x&y \end{array} \right][/latex] [latex]\left[ \begin{array}{ccc} a&b&c\\d&e&f\\k&m&n \end{array} \right][/latex] [latex]\left[ \begin{array}{c} x\\y\\1 \end{array} \right][/latex] has a name?There are three matrices there. Three. THREE. Edited November 2, 2011 by the tree Link to comment Share on other sites More sharing options...
mississippichem Posted November 2, 2011 Share Posted November 2, 2011 I usually call them Jim, John, and Terry. Terry is recently divorced though and doesn't feel like talking much anymore. 1 Link to comment Share on other sites More sharing options...
imatfaal Posted November 2, 2011 Share Posted November 2, 2011 I usually call them Jim, John, and Terry. Terry is recently divorced though and doesn't feel like talking much anymore. Quick someone get to Mississippi's lab - he has been sniffing the wrong chemicals again 1 Link to comment Share on other sites More sharing options...
Mellinia Posted November 3, 2011 Author Share Posted November 3, 2011 There are three matrices there. Three. THREE. Sorry. It should be matrices. So it doesn't, collectively, have a name? I can see that it represents the equation of a curve. Link to comment Share on other sites More sharing options...
the tree Posted November 3, 2011 Share Posted November 3, 2011 (edited) It doesn't really represent anything. Are those supposed to be multiplied together? Are we talking dot multiplication? Even that only makes sense if there are the same amount of rows on the left as columns on the right of the dot. edit Okay, it occurs to me that you might have meant [math]\left[ \begin{array}{ccc} x&y&1 \end{array} \right] \cdot \left[ \begin{array}{ccc} a&b&c\\d&e&f\\k&m&n \end{array} \right] \cdot \left[ \begin{array}{c} x\\y\\1 \end{array} \right] [/math] Which does indeed expand to the expression in the OP. In which case, no name, just a very obscure and typo ridden way of writing the general form of a two variable polynomial of degree two. It doesn't represent an equation. It's an expression. Equations have equals in them. Edited November 3, 2011 by the tree Link to comment Share on other sites More sharing options...
imatfaal Posted November 3, 2011 Share Posted November 3, 2011 Set equal to zero and with certain coefficients you will obtain an ellipse - other forms will give hyperbola or no real plots Link to comment Share on other sites More sharing options...
the tree Posted November 3, 2011 Share Posted November 3, 2011 I think it covers all the conic sections, although I really can't see any merit to writing them like that. Link to comment Share on other sites More sharing options...
imatfaal Posted November 3, 2011 Share Posted November 3, 2011 I think it covers all the conic sections, although I really can't see any merit to writing them like that. yes of course. should have thought - if the coefficient for the y^2 and xyare zero it will be a parabola. But agree it is a very strange way of writing it Link to comment Share on other sites More sharing options...
the tree Posted November 3, 2011 Share Posted November 3, 2011 It turns out this is actually a thing, and you can sort of get something from it. Still no name for it though. Link to comment Share on other sites More sharing options...
Mellinia Posted November 4, 2011 Author Share Posted November 4, 2011 It doesn't really represent anything. Are those supposed to be multiplied together? Are we talking dot multiplication? Even that only makes sense if there are the same amount of rows on the left as columns on the right of the dot. edit Okay, it occurs to me that you might have meant [math]\left[ \begin{array}{ccc} x&y&1 \end{array} \right] \cdot \left[ \begin{array}{ccc} a&b&c\\d&e&f\\k&m&n \end{array} \right] \cdot \left[ \begin{array}{c} x\\y\\1 \end{array} \right] [/math] Which does indeed expand to the expression in the OP. In which case, no name, just a very obscure and typo ridden way of writing the general form of a two variable polynomial of degree two. It doesn't represent an equation. It's an expression. Equations have equals in them. Yes I did. Sorry I missed out an element. So basically it can be used to represent various conics...but I see that the normal way is much simpler and less fussier. I actually done a differentiation operation of it by expanding it but I don't think it has much use anyway. Link to comment Share on other sites More sharing options...
the tree Posted November 4, 2011 Share Posted November 4, 2011 If you take a look at the wikipedia article I linked, you'll see that by taking determinants of the matrix in the middle and of a minor of that matrix, you can see what kind of conic you are looking at. Link to comment Share on other sites More sharing options...
Mellinia Posted November 5, 2011 Author Share Posted November 5, 2011 We can do the same by determining the eccentricity of the conic using the conventional method...but the matrix method conveys it much easily Link to comment Share on other sites More sharing options...
DrRocket Posted November 5, 2011 Share Posted November 5, 2011 Yes I did. Sorry I missed out an element. So basically it can be used to represent various conics...but I see that the normal way is much simpler and less fussier. I actually done a differentiation operation of it by expanding it but I don't think it has much use anyway. What you have is the restriction of a quadratic form on 3-space to the two-dimensional subspace determined by the variables x and y. This is a rather unusual notion. More commonly one studies quadratic forms of some given dimension which are determined by a symmetric matrix, often positive-definite, and almost always non-degenerate (determinant not 0). There is a large literature on quadratic forms -- see for instance the books by O'Meara or Lam. Link to comment Share on other sites More sharing options...
Mellinia Posted November 6, 2011 Author Share Posted November 6, 2011 What you have is the restriction of a quadratic form on 3-space to the two-dimensional subspace determined by the variables x and y. This is a rather unusual notion. Quadratic forms are expressions of space? Link to comment Share on other sites More sharing options...
DrRocket Posted November 6, 2011 Share Posted November 6, 2011 (edited) Quadratic forms are expressions of space? No. Quadratic forms are functions defined on vectors from an n-dimensional vector space. I have no idea what an "expression of space" could possibly mean. Edited November 6, 2011 by DrRocket Link to comment Share on other sites More sharing options...
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