Comments on: What is the cosine of a matrix? http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/ The blog of John D. Cook Sat, 11 Feb 2012 01:10:06 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: John http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/comment-page-1/#comment-130938 John Mon, 16 Jan 2012 17:02:18 +0000 http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/#comment-130938 Jordan canonical form is useful in exact hand calculations with small or special matrices. It's unsuitable for numerical calculation because it is discontinuous: the tiniest change to a matrix can change a 0 to a 1 in the JCF. I imagine power series could be practical, though not always a power series centered at 0. You probably want to start with a nearby matrix that is easy to exponentiate. Finding the cosine of A is equivalent to solving a system of differential equations. It may be better numerically to solve the differential equations directly. Jordan canonical form is useful in exact hand calculations with small or special matrices. It’s unsuitable for numerical calculation because it is discontinuous: the tiniest change to a matrix can change a 0 to a 1 in the JCF.

I imagine power series could be practical, though not always a power series centered at 0. You probably want to start with a nearby matrix that is easy to exponentiate.

Finding the cosine of A is equivalent to solving a system of differential equations. It may be better numerically to solve the differential equations directly.

]]> By: nils http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/comment-page-1/#comment-130934 nils Mon, 16 Jan 2012 16:53:19 +0000 http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/#comment-130934 Is the power series the best way to calculate cos(A) or are there other ways? In that case, which are they? Jordanization? Is the power series the best way to calculate cos(A) or are there other ways? In that case, which are they? Jordanization?

]]> By: nils http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/comment-page-1/#comment-130926 nils Mon, 16 Jan 2012 16:31:03 +0000 http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/#comment-130926 Is the power series the best way to calculate cos(A) or are there other ways? In that case, which are they? Is the power series the best way to calculate cos(A) or are there other ways? In that case, which are they?

]]> By: John http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/comment-page-1/#comment-115715 John Fri, 18 Nov 2011 13:50:24 +0000 http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/#comment-115715 human mathematics: because inverse cosine is the angle between two unit vectors. See <a href="http://www.johndcook.com/blog/2010/06/17/covariance-and-law-of-cosines/" rel="nofollow">covariance and cosines</a>. human mathematics: because inverse cosine is the angle between two unit vectors. See covariance and cosines.

]]> By: human mathematics http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/comment-page-1/#comment-115714 human mathematics Fri, 18 Nov 2011 13:46:36 +0000 http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/#comment-115714 Another application is in recommender systems. Cosine distance is one of the ways to compare my library to yours. Another application is in recommender systems. Cosine distance is one of the ways to compare my library to yours.

]]> By: Ben N http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/comment-page-1/#comment-41750 Ben N Thu, 15 Jul 2010 02:49:26 +0000 http://www.johndcook.com/blog/2008/03/14/what-is-the-cosine-of-a-matrix/#comment-41750 Actually, (A+B)<sup>2</sup> = A<sup>2</sup> + AB + BA + B<sup>2</sup> whether or not A commutes with B, although if they don't commute this won't equal A<sup>2 </sup>+ 2AB + B<sup>2</sup>. The issue here is one of <i>starting</i> with (A+B)<sup>n</sup>, for all n, and being unable to group terms. Instead of getting a term for each combination of A and B, you get a term for each <i>permutation</i> and there's just nothing sensible to be done with them that will bear any resemblance to the familiar world of commutative algebra. Actually, (A+B)2 = A2 + AB + BA + B2 whether or not A commutes with B, although if they don’t commute this won’t equal A2 + 2AB + B2. The issue here is one of starting with (A+B)n, for all n, and being unable to group terms. Instead of getting a term for each combination of A and B, you get a term for each permutation and there’s just nothing sensible to be done with them that will bear any resemblance to the familiar world of commutative algebra.

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