Power of two - Revision history

93.173.10.175: /* Fast algorithm to check if a positive number is a power of two */ A better wikilink for "bits".

2015-01-03T11:20:18Z

Fast algorithm to check if a positive number is a power of two: A better wikilink for "bits".

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en>JordanKyser22: /* The first 96 powers of two */

2014-03-02T19:59:20Z

The first 96 powers of two

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217.24.251.208: /* Powers of 1024 */

2014-01-16T00:49:37Z

Powers of 1024

← Older revision		Revision as of 02:49, 16 January 2014
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			:<math>\begin{bmatrix}
			a & b & c & d & e \\
			b & c & d & e & f \\
			c & d & e & f & g \\
			d & e & f & g & h \\
			e & f & g & h & i \\
			\end{bmatrix}.</math>

			If the ''i'',''j'' element of ''A'' is denoted ''A''<sub>''i'',''j''</sub>, then we have

			:<math>A_{i,j} = A_{i-1,j+1}.\ </math>

			The Hankel matrix is closely related to the [[Toeplitz matrix]] (a Hankel matrix is an upside-down Toeplitz matrix). For a special case of this matrix see [[Hilbert matrix]].

			A Hankel [[operator (mathematics)\|operator]] on a [[Hilbert space]] is one whose matrix with respect to an [[orthonormal basis]] is a (possibly infinite) Hankel matrix
			<math>(A_{i,j})_{i,j \ge 1}</math>, where <math> A_{i,j}</math> depends only on <math>i+j</math>.

			The determinant of a Hankel matrix is called a [[catalecticant]].

			==Hankel transform==
			The '''Hankel transform''' is the name sometimes given to the transformation of a [[sequence]], where the transformed sequence corresponds to the determinant of the Hankel matrix. That is, the sequence <math>\{h_n\}_{n\ge 0}</math> is the Hankel transform of the sequence <math>\{b_n\}_{n\ge 0}</math> when

			:<math>h_n = \det (b_{i+j-2})_{1 \le i,j \le n+1}.</math>

			Here, <math>a_{i,j}=b_{i+j-2}</math> is the Hankel matrix of the sequence <math>\{b_n\}</math>. The Hankel transform is invariant under the [[binomial transform]] of a sequence. That is, if one writes

			:<math>c_n = \sum_{k=0}^n {n \choose k} b_k</math>

			as the binomial transform of the sequence <math>\{b_n\}</math>, then one has

			:<math>\det (b_{i+j-2})_{1 \le i,j \le n+1} = \det (c_{i+j-2})_{1 \le i,j \le n+1}.</math>

			== Hankel matrices for system identification ==
			Hankel matrices are formed when given a sequence of output data and a realization of an underlying state-space or hidden Markov model is desired. The [[singular value decomposition]] of the Hankel matrix provides a means of computing the A, B, and C matrices which define the state-space realization.

			== Orthogonal polynomials on the real line ==
			{{Incomplete\|section\|date=February 2009}}

			=== Positive Hankel matrices and the Hamburger moment problems ===
			{{further\|Hamburger moment problem}}

			=== Orthogonal polynomials on the real line ===

			=== Tridiagonal model of positive Hankel operators ===

			==See also==
			* [[Hamburger moment problem]]
			* [[Toeplitz matrix]], a Hankel matrix 'upside-down'.

			==References==
			* {{cite book \| title=An introduction to Hankel operators \| author=J.R. Partington \| authorlink=Jonathan Partington \| series=LMS Student Texts \| volume=13 \| publisher=[[Cambridge University Press]] \| year=1988 \| isbn=0-521-36791-3 }}

			[[Category:Matrices]]
			[[Category:Transforms]]


			{{Linear-algebra-stub}}

174.98.8.39: /* The first 96 powers of two */

2012-09-02T03:44:24Z

The first 96 powers of two

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