Complex vector inner products: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
Line 4: | Line 4: | ||
<math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^n u_k^* v_k</math> |
<math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^n u_k^* v_k</math> |
||
where we have assumed that the basis vectors are orthonormal and that there are <math>n</math> dimensions. Some define the conjugate on the second vector instead of the first, but in order to be consistent with bra-ket notation from quantum mechanics, we do it with the conjugate on the first vector. |
where we have assumed that the basis vectors are orthonormal and that there are <math>n</math> dimensions. The <math> u_k^* </math> is the [http://en.wikipedia.org/wiki/Complex_conjugate complex conjugate] of <math> u_k </math>. Some define the conjugate on the second vector instead of the first, but in order to be consistent with bra-ket notation from quantum mechanics, we do it with the conjugate on the first vector. |
||
With this we have |
With this we have |
Revision as of 16:43, 26 September 2004
Complex Vector Inner Products
In order to preserve the property that the inner product of any vector with itself is the magnitude of that vector squared, we need to define the inner product of complex vectors so that
where we have assumed that the basis vectors are orthonormal and that there are dimensions. The is the complex conjugate of . Some define the conjugate on the second vector instead of the first, but in order to be consistent with bra-ket notation from quantum mechanics, we do it with the conjugate on the first vector.
With this we have
and
.
Principle author of this page: Rob Frohne