Difference between revisions of "Unitary matrix"
From Encyclopedia of Mathematics
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | u0955401.png | ||
| + | $#A+1 = 11 n = 0 | ||
| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/U095/U.0905540 Unitary matrix | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| − | + | {{TEX|auto}} | |
| + | {{TEX|done}} | ||
| − | + | A square matrix $ A = \| a _ {ik} \| _ {1} ^ {n} $ | |
| + | over the field $ \mathbf C $ | ||
| + | of complex numbers, whose rows form an orthonormal system, i.e. | ||
| − | + | $$ | |
| + | a _ {i1} \overline{a}\; _ {k1} + \dots + a _ {in} \overline{a}\; _ {kn} = \ | ||
| + | \left \{ | ||
| − | + | $ i, k = 1 \dots n $. | |
| + | In a [[Unitary space|unitary space]], transformation from one orthonormal basis to another is accomplished by a unitary matrix. The matrix of a [[Unitary transformation|unitary transformation]] relative to an orthonormal basis is also (called) a unitary matrix. A square matrix $ A $ | ||
| + | with complex entries is unitary if and only if it satisfies any of the following conditions: | ||
| − | + | 1) $ A ^ {*} A = E $; | |
| − | + | 2) $ AA ^ {*} = E $; | |
| − | + | 3) $ A ^ {*} = A ^ {-} 1 $; | |
| + | 4) the columns of $ A $ | ||
| + | form an orthonormal system (here $ A ^ {*} $ | ||
| + | is the conjugate transposed of $ A $). | ||
| + | The determinant of a unitary matrix is a complex number of modulus one. | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. 63</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.H. Greub, "Linear algebra" , Springer (1975) pp. 329</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. 63</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.H. Greub, "Linear algebra" , Springer (1975) pp. 329</TD></TR></table> | ||
Latest revision as of 08:27, 6 June 2020
A square matrix $ A = \| a _ {ik} \| _ {1} ^ {n} $
over the field $ \mathbf C $
of complex numbers, whose rows form an orthonormal system, i.e.
$$ a _ {i1} \overline{a}\; _ {k1} + \dots + a _ {in} \overline{a}\; _ {kn} = \ \left \{
$ i, k = 1 \dots n $. In a unitary space, transformation from one orthonormal basis to another is accomplished by a unitary matrix. The matrix of a unitary transformation relative to an orthonormal basis is also (called) a unitary matrix. A square matrix $ A $ with complex entries is unitary if and only if it satisfies any of the following conditions:
1) $ A ^ {*} A = E $;
2) $ AA ^ {*} = E $;
3) $ A ^ {*} = A ^ {-} 1 $;
4) the columns of $ A $ form an orthonormal system (here $ A ^ {*} $ is the conjugate transposed of $ A $).
The determinant of a unitary matrix is a complex number of modulus one.
Comments
References
| [a1] | W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. 63 |
| [a2] | W.H. Greub, "Linear algebra" , Springer (1975) pp. 329 |
How to Cite This Entry:
Unitary matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_matrix&oldid=49083
Unitary matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_matrix&oldid=49083
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article