en>LokiClock: /* Intro */ Nevermind, specialized sense

2013-03-31T06:18:38Z

Intro: Nevermind, specialized sense

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			In [[linear algebra]] ([[mathematics]]), the '''principal angles''', also called '''canonical angles''', provide information about the relative position of two [[Linear subspace\|subspaces]] of an [[inner product space]]. The concept was first introduced by [[Camille Jordan\|Jordan]] in 1875.

			==Definition==
			Let <math>V</math> be an inner product space.
			Given two subspaces <math>\mathcal{U},\mathcal{W}</math> with
			<math>\operatorname{dim}(\mathcal{U})=k\leq \operatorname{dim}(\mathcal{W}):=l</math>,
			there exists then a sequence of <math>k</math> angles
			<math> 0 \le \theta_1 \le \theta_2 \le \ldots \le \theta_k \le \pi/2</math>
			called the principal angles, the first one defined as

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			where <math>\langle \cdot , \cdot \rangle </math> is the [[inner product]] and <math>\\|\cdot\\|</math> the induced [[Norm_(mathematics)\|norm]]. The vectors <math>u_1</math> and <math>w_1</math> are the corresponding ''principal vectors.''

			The other principal angles and vectors are then defined recursively via

			: <math>\theta_i:=\min \left\{ \left. \arccos \left( \frac{ \|\langle u,w\rangle\| }{\\|u\\| \\|w\\|}\right) \right\| u\in \mathcal{U},~w\in \mathcal{W},~u\perp u_j,~w \perp w_j \quad \forall j\in \{1,\ldots,i-1\} \right\}.</math>

			This means that the principal angles
			<math>(\theta_1,\ldots \theta_k)</math>
			form a set of minimized angles between the two subspaces, and the principal vectors in each subspace are orthogonal to each other.

			==Examples==
			===Geometric Example===
			Geometrically, subspaces are [[Flat (geometry)\| flats]] (points, lines, planes etc.) that include the origin, thus any two subspaces intersect at least in the origin. Two two-dimensional subspaces <math>\mathcal{U}</math> and <math>\mathcal{W}</math> generate a set of two angles. In a three-dimensional [[Euclidean space]], the subspaces <math>\mathcal{U}</math> and <math>\mathcal{W}</math> are either identical, or their intersection forms a line. In the former case, both <math>\theta_1=\theta_2=0</math>. In the latter case, only <math>\theta_1=0</math>, where vectors <math>u_1</math> and <math>w_1</math> are on the line of the intersection <math>\mathcal{U}\cap\mathcal{W}</math> and have the same direction. The angle <math>\theta_2>0</math> will be the angle between the subspaces <math>\mathcal{U}</math> and <math>\mathcal{W}</math> in the [[orthogonal complement]] to <math>\mathcal{U}\cap\mathcal{W}</math>. Imagining the angle between two planes in 3D, one intuitively thinks of the largest angle, <math>\theta_2>0</math>.

			===Algebraic Example===
			In 4-dimensional real coordinate space '''R'''<sup>''4''</sup>, let the two-dimensional subspace <math>\mathcal{U}</math> be
			spanned by <math>u_1=(1,0,0,0)</math> and <math>u_2=(0,1,0,0)</math>, while the two-dimensional subspace <math>\mathcal{W}</math> be
			spanned by <math>w_1=(1,0,0,a)/\sqrt{1+a^2}</math> and <math>w_2=(0,1,b,0)/\sqrt{1+b^2}</math> with some real <math>a</math> and <math>b</math> such that <math>\|a\|<\|b\|</math>. Then <math>u_1</math> and <math>w_1</math> are, in fact, the pair of principal vectors corresponding to the angle <math>\theta_1</math> with <math>\cos(\theta_1)=1/\sqrt{1+a^2}</math>, and <math>u_2</math> and <math>w_2</math> are the principal vectors corresponding to the angle <math>\theta_2</math> with <math>\cos(\theta_2)=1/\sqrt{1+b^2}</math>

			To construct a pair of subspaces with any given set of <math>k</math> angles <math>\theta_1,\ldots,\theta_k</math> in a <math>2k</math> (or larger) dimensional [[Euclidean space]], take a subspace <math>\mathcal{U}</math> with an orthonormal basis <math>(e_1,\ldots,e_k)</math> and complete it to an orthonormal basis <math>(e_1,\ldots, e_n)</math> of the Euclidean space, where <math>n\geq 2k</math>. Then, an orthonormal basis of the other subspace <math>\mathcal{W}</math> is, e.g.,

			: <math>(\cos(\theta_1)e_1+\sin(\theta_1)e_{k+1},\ldots,\cos(\theta_k)e_k+\sin(\theta_k)e_{2k}).</math>

			==Basic Properties==
			If the largest angle is zero, one subspace is a subset of the other.

			If the smallest angle is zero, the subspaces intersect at least in a line.

			The number of angles equal to zero is the dimension of the space where the two subspaces intersect.

			==References==
			*[http://www.ams.org/bull/1902-08-08/S0002-9904-1902-00903-8/S0002-9904-1902-00903-8.pdf Concerning the angles and the angular determination] English lecture dealing with the original work of Jordan read by Keyeser, 1902.
			*[http://www.haverford.edu/math/cshonkwi/research/notes/PrincipalAngles.pdf Principal angles in terms of inner product] Downloadable note on the construction of principal angles, Shonkwiler, Haverford

			[[Category:Linear algebra]]

en>Linas: Category:Free algebraic structures

2012-08-25T16:42:31Z

Category:Free algebraic structures

New page

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en>LokiClock: /* Intro */ Nevermind, specialized sense

en>Linas: Category:Free algebraic structures