Wikipedia:Reference desk/Archives/Mathematics/2009 May 18

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May 18

Decomposition of a symmetric matrix

Hi there - I'm trying to show that for any non-zero vector vi , any 3×3 symmetric matrix T_ij can be expressed as $T_{ij}=A\delta _{ij}+Bv_{i}v_{j}+(C_{i}v_{j}+C_{j}v_{i})+D_{ij}$ for some numbers A and B, some vector C_i and a symmetric matrix D_ij , where $C_{i}v_{i}=0,D_{ii}=0,D_{ij}v_{j}=0$. (Summation convention implicit here).

I've been told to show that the above statement is true by finding A, B, C_i and D_ij explicitly in terms of T and v - and I've been fiddling around for a while now, not achieved much more than $C_{i}={\frac {T_{ij}v_{j}}{T_{ii}}}$ and a few expressions in A/B/etc by dotting T with c and v. Can anyone suggest where I might go next? 131.111.8.102 (talk) 08:36, 18 May 2009 (UTC)

I guess that the best way should be to understand geometrically the problem; however, while waiting for a deeper answer from other people, here are at least some remarks. I'm using the same summation convention.

1. In order to simplify notations you can also assume that $v:=(v_{1},v_{2},v_{3})$ is a unit vector, that is $v_{i}v_{i}=1$. For a general non-zero $v$, just normalize it.

2. If there is a solution, it is unique; in particular, we find from the equations:

(2.1) $\scriptstyle T_{ij}v_{i}v_{j}=A+B$;

(2.2) $\scriptstyle T_{ij}v_{j}=Av_{i}+Bv_{i}+C_{i}=(A+B)v_{i}+C_{i}$;

(2.3) $\scriptstyle T_{ii}=3A+B$;

so, equivalently

$\scriptstyle A={\big (}T_{ii}-T_{ij}v_{i}v_{j}{\big )}/2;$

$\scriptstyle B={\big (}3T_{ij}v_{i}v_{j}-T{ii}{\big )}/2;$

$\scriptstyle C_{i}=T_{ij}v_{j}-T_{kj}v_{k}v_{j}v_{i}$

(the last seems to differ from your expression of C_i: how did you get it?). Hence you also have $D_{ij}$ by the initial equation. Thus, if there exists that decomposition, it is unique, and the value of $A,B,C_{i},D_{ij}$ are the one we found. Last step, you should plug these value into the equations and verify that they actually solve the problem (this seems OK to me, but you better check it). --pma (talk) 14:46, 18 May 2009 (UTC)

Addendum: in fact, you have the analogue unique decomposition of a symmetric matrix $T$ in any dimension $n>1$; in eq (2.3) you only have an $n$ instead of 3. For the computation you might find it easier (as I do) to use a matrix notation. You want numbers $A$ and $B$, a vector $C$ and a symmetric matrix $D$, such that for all vectors $x$ there hold:

$\scriptstyle Tx=Ax+B(v\cdot x)v+(C\cdot x)v+(v\cdot x)C+Dx;$

$\scriptstyle (C\cdot v)=0;$

$\scriptstyle Dv=0;$

$\scriptstyle \mathrm {tr} (D)=0.$

Assuming as before $\scriptstyle \|v\|=1$, the unique solution is given by

$\scriptstyle A:={\frac {1}{n-1}}{\big (}\mathrm {tr} (T)-(Tv\cdot v){\big )}$

$\scriptstyle B:={\frac {1}{n-1}}{\big (}n(Tv\cdot v)-\mathrm {tr} (T){\big )}$

$\scriptstyle C:=Tv-(Tv\cdot v)v$

$\scriptstyle Dx:=Tx-Ax-B(v\cdot x)v-(C\cdot x)v-(v\cdot x)C.$

Ask for further details if you need it. --pma (talk) 17:35, 18 May 2009 (UTC)

I re-approached the question and noticed an error in my C - thankfully I concur with you! The question also claims the A/B/C/D space is exactly the correct dimension to parametrize an arbitrary 3x3 symmetric matrix - why is that? I can't see a way to justify this - why would the space created by A-B-C-D be of the correct size? I can see how the fact that any such matrix has a single exact decomposition would imply that the space uniquely defines a map so I can see -how- it's the correct size, but I can't actually see -why-: what's the reason behind this? Does anyone have any insight, algebraic or geometric or any? Thanks!

Oh, and thankyou very much for the help pma, I was just making a big mess of things because I didn't spot an easy way to obtain 1 last equation so I was trying everything complicated, not very carefully apparently! I guess I just need to be more cautious when solving these sorts of things - thanks very much for the help again, 131.111.8.96 (talk) 18:30, 18 May 2009 (UTC)

At least, a dimensional check is easy. First, A and B are constants and C is an n-vector orthogonal to v: hence the (A,B,C)-space has dimension 1+1+(n-1)=n+1. What is the dimension of the space of all n×n symmetric matrices D with null trace and with the non-zero vector v in the kernel? If you observe that for an orthogonal matrix U the matrix UDU^T is still symmetric, with null trace, and with Uv in the kernel, you should realize that one can assume v=e₁=(1,0,..,0) with no loss of generality. In this case it is evident that D is fixed by n(n+1)/2 coefficients with n+1 independent linear conditions (all entries in the first column together with the trace of D are zero). So the D-space has dimension n(n+1)/2-(n+1)=(n+1)(n-2)/2, and the (A,B,C,D)-space has the right dimension n(n+1)/2 of the space of n×n symmetric matrices. In other words, with the assumption v=e₁ (that you can have after an orthogonal change of basis as said above), the four terms decomposition of the symmetric matrix T is just the sum of: a multiple of the identity; a (symmetric) matrix with support in the corner (1,1) that is, whose coefficients are zero except possibly the one of index (1,1) ; a symmetric matrix with support in the first frame (that is, whose coefficients are zero except possibly those in the first column or in the first row); a symmetric matrix with zero trace and support in the lower (n-1)×(n-1) square (that is, the first row and the first column are zero). --pma (talk) 21:16, 18 May 2009 (UTC)

how many spheres diam D1 can pack on the surface of a larger sphere diam D2

If you have spheres of diam D1, and a sphere of diam D2, D1< D2, how many small spheres can "pack" on the surface of the larger sphere ? If you have an answer, you might consider adding it to the article "sphere" thanks Cinnamon colbert (talk) 12:52, 18 May 2009 (UTC)

Please expand this question. A sphere is 3D, the surface of a sphere is 2D. I can imagine asking the question "How many circles can one fit on the surface of a sphere?", but what does packing spheres on the surface mean? In the event that you mean circles on a sphere's surface, what are you defining as the radius of the circles: The great circle distance, the circle centre to the circumference linear distance, the distance across a "2d chord" to the centre of a 2D circle whose centre is not on the sphere's surface, some other option...? -- SGBailey (talk) 14:30, 18 May 2009 (UTC)

sorry; if you have a sphere (say a basketball) you can put a pingpong ball on the surface, so the two are touching. The question is, how many pingpong balls can touch the surface of the basketball - this is an extension of the kissing number ( http://local.wasp.uwa.edu.au/~pbourke/geometry/kissing/ ) which addresses spheres of the same size —Preceding unsigned comment added by Cinnamon colbert (talk • contribs) 16:46, 18 May 2009 (UTC)

The problem is usually stated in terms of disks: the circle that is "under" the pingpong ball with respect to the basketball's gravity.

No general answer is known (though an upper bound is easy). For some small n, proofs exist that a specific arrangement is the densest packing of n disks on a sphere, and thus for bigger disks the packing-number is less than n; for all other n, all we have is best known packings. Some solutions are linked at [1]. —Tamfang (talk) 16:59, 18 May 2009 (UTC)

thanks, tamfang, that looks great, cinn col —Preceding unsigned comment added by 65.220.64.105 (talk) 17:45, 18 May 2009 (UTC)

On a similar note, I've been trying to show topologically (looking at you if you're bored PST! ;)) that in $\mathbb {R} ^{n}$ there exist $(1+c)^{n}$ disjoint closed unit balls inside any closed ball of radius 3.001 (or an analogous ratio - does the 3.001 specifically matter?) for some constant c > 0 (using the Euclidean metric) - can anyone suggest a starting point? Cheers muchly 131.111.8.96 (talk) 18:44, 18 May 2009 (UTC)

Try seeing what happens when you double the size of each of the sphheres, that should cover every single point inside the large sphere possibly a number of times for some points. What is the ratio of the volume of all the spheres to the large sphere? This seems to indicate that just over 2 for the large sphere should be good enough but it seems rather counter intuitive to me so I'd check the logic carefully. Dmcq (talk) 09:54, 19 May 2009 (UTC)

This was wrong, see WP:RD/MA#Spheres in an n-dimensional sphere below. If the big sphere has diameter greater than 1+sqrt(2) then the number goes up as a power of n. For that number it just goes up linearly. Dmcq (talk) 11:57, 22 May 2009 (UTC)

Still no simple answer to what i thought would be a simple question: if you put a small sphere onto the surface of a larger sphere (in the commonsense usage, onto just touching) is, given the diameters of the 2 spheres, there a simple formula that says how many small spheres can coat the surface of the larger sphere . I don't need a guaranteed math best result; this is more of an engineering type question (and maybe I should go there) - I'm happy with a formula that is within ~90% of bestCinnamon colbert (talk) 13:50, 19 May 2009 (UTC)

If D1<<D2, a reasonable estimation should be: the number N of 2 dimensional disks of diameter D1 that you can pack in a disk of diameter 2(D1+D2), which has the same surface area of a sphere of diameter D1+D2: thus, N≈4c((D2/D1)+1)², where c=0.906.. is the highest density for 2 dimensional unit disks. The idea is that we are looking at the traces of the small balls on the sphere of diameter D2+D1, like it were a planar disk packing.--131.114.72.215 (talk) 14:06, 19 May 2009 (UTC)

Consider the spherical triangle formed by the centers of three discs, all tangent to each other. That triangle contains 1/6 of each of three discs. Using spherical trigonometry, find the angle α of the triangle, which will exceed π/3. If my algebra is right, the number of such discs that will fit on a sphere is no more than ${\frac {2\pi }{3\alpha -\pi }}$. —Tamfang (talk) 06:30, 20 May 2009 (UTC)

Oops, wrong wrong wrong. —Tamfang (talk) 14:47, 20 May 2009 (UTC)

I think I have it right this time. Let r be the angular radius of a disc; β = asin((√3/2) tan r); the number of discs ≤ 12β/(6β-π). —Tamfang (talk) 15:13, 20 May 2009 (UTC)

What is the ratio of the area of a circle to a square

What is the ratio of the area of a circle to a square?

Why is answer arctan(1/2) + arctan(1/3) ? 122.107.207.98 (talk) 13:23, 18 May 2009 (UTC)

Actually, the answer is 5, or 17, or any other number. It really depends on the relative size of the two. Is this homework? --Stephan Schulz (talk) 14:00, 18 May 2009 (UTC)

Since the answer is supposed to be arctan(1/2) + arctan(1/3) = π/4, we can deduce that it is a ratio of the area of a circle with radius r to a square with side 2r. — Emil J. 14:24, 18 May 2009 (UTC)

So the ratio of the area of a square to the area of an inscribed circle. --Tango (talk) 18:13, 18 May 2009 (UTC)

Please, people. When someone posts what is clearly a homework answer, you should not produce the question for them. 79.122.45.200 (talk) 18:29, 18 May 2009 (UTC)

Saying that the answer is arctan(1/2) + arctan(1/3) is certainly not a good hint at how to answer the initial question (unless maybe the problem was approached via some particular geometric argument that you haven't told us about). It's really quite a separate question. Recall the formula for the tangent of a sum:

$\tan(\alpha +\beta )={\frac {\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }},$

and as a consequence

$\arctan x+\arctan y=\arctan {\frac {x+y}{1-xy}}$

(except that here you have to qualify the statement because of the "multiple-valued" nature of inverse trigonometric functions).

So plug arctan(1/2) + arctan(1/3) into that formula and by the time you're done simplifying you've got something really simple.

As for the ratio of areas, you should know how to find the area of a circle and how to find the area of a square (if you know the radius of the circle and the length of the side of a square). Michael Hardy (talk) 18:59, 18 May 2009 (UTC)

90 Degree Angle

Is a 90 degree angle acute, obtuse, or fall into its own category, as a right angle? I know that anything bigger that 90 is obtuse, and anything smaller is acute, but does that mean that a 90 degree angle is a right angle? —Preceding unsigned comment added by 99.58.205.224 (talk) 23:01, 18 May 2009 (UTC)

Yes, 90 degrees is called a right angle. That is a separate category inbetween acute and obtuse. --Tango (talk) 23:30, 18 May 2009 (UTC)

Ah, the acute, right and obtuse angles!! Great times, when I heard about this stuff for the first time. What I found exciting about maths at that age was that everything we learned at school, we would find there out, in the same day. --pma (talk) 06:13, 19 May 2009 (UTC)

As a maths teacher I get the young students to count how many right angles they can see from where they are sitting. Only then do I throw Pythagoras and the 3-4-5 triangle at them. Cuddlyable3 (talk) 11:04, 19 May 2009 (UTC)