Algor mortis: Difference between revisions

Revision as of 22:14, 29 January 2014

I'm Fernando (21) from Seltjarnarnes, Iceland.
I'm learning Norwegian literature at a local college and I'm just about to graduate.
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my site; wellness [continue reading this..] In mathematics, a unitary transformation may be informally defined as a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.

More precisely, a unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function

U : H_{1} \to H_{2}

where $H_{1}$ and $H_{2}$ are Hilbert spaces, such that

⟨ U x, U y ⟩ = ⟨ x, y ⟩

for all $x$ and $y$ in $H_{1}$ . A unitary transformation is an isometry, as one can see by setting $x = y$ in this formula.

In the case when $H_{1}$ and $H_{2}$ are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

A closely related notion is that of antiunitary transformation, which is a bijective function

U : H_{1} \to H_{2}

between two complex Hilbert spaces such that

⟨ U x, U y ⟩ = \overline{⟨ x, y ⟩} = ⟨ y, x ⟩

for all $x$ and $y$ in $H_{1}$ , where the horizontal bar represents the complex conjugate.

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+{{other uses|Transformation (mathematics) (disambiguation)}}
+In mathematics, a '''unitary transformation''' may be informally defined as a [[transformation (mathematics)|transformation]] that preserves the [[inner product]]: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
+More precisely, a '''unitary transformation''' is an [[isomorphism]] between two [[Hilbert space]]s. In other words, a ''unitary transformation'' is a [[bijective function]]
+:<math>U:H_1\to H_2\,</math>
+where <math>H_1</math> and <math>H_2</math> are Hilbert spaces, such that
+:<math>\langle Ux, Uy \rangle = \langle x, y \rangle</math>
+for all <math>x</math> and <math>y</math> in <math>H_1</math>. A unitary transformation is an [[isometry]], as one can see by setting <math>x=y</math> in this formula.
+In the case when <math>H_1</math> and <math>H_2</math> are the same space, a unitary transformation is an [[automorphism]] of that Hilbert space, and then it is also called a [[unitary operator]].
+A closely related notion is that of '''[[antiunitary]] transformation''', which is a bijective function
+:<math>U:H_1\to H_2\,</math>
+between two [[complex number|complex]] Hilbert spaces such that
+:<math>\langle Ux, Uy \rangle = \overline{\langle x, y \rangle}=\langle y, x \rangle</math>
+for all <math>x</math> and <math>y</math> in <math>H_1</math>, where the horizontal bar represents the [[complex conjugate]].
+==See also==
+*[[Antiunitary]]
+*[[Orthogonal transformation]]
+*[[Time reversal]]
+*[[Unitary group]]
+*[[Unitary operator]]
+*[[Unitary matrix]]
+*[[Wigner's Theorem]]
+{{Mathanalysis-stub}}
+[[Category:Linear algebra]]
+[[Category:Functional analysis]]
+[[ru:Унитарное преобразование]]

Algor mortis: Difference between revisions

Revision as of 22:14, 29 January 2014

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