B | = | |B > | = |
┌ b_{1} ┐ │ b_{2} │ └ b_{3} ┘ |
<A|B> | = | [a_{1},a_{2},a_{3}] | · |
┌ b_{1} ┐ │ b_{2} │ └ b_{3} ┘ | = | a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} |
┌ a_{1}^{*} ┐ │ a_{2}^{*} │ └ a_{3}^{*} ┘ |
┌ a_{11} a_{12} a_{13} ┐ │ a_{21} a_{22} a_{23} │ └ a_{31} a_{32} a_{33}┘ |
┌ B_{1} ┐ │ B_{2} │ └ B_{3} ┘ |
= |
┌ a_{11}*B_{1}+a_{12}*B_{2}+a_{13}*B_{3} ┐ │ a_{21}*B_{1}+a_{22}*B_{2}+a_{23}*B_{3} │ └ a_{31}*B_{1}+a_{32}*B_{2}+a_{33}*B_{3} ┘ |
= |
┌ C_{1} ┐ │ C_{2} │ └ C_{3} ┘ |
[ A_{1} A_{2} A_{3} ] |
┌ a_{11}*B_{1}+a_{12}*B_{2}+a_{13}*B_{3} ┐ │ a_{21}*B_{1}+a_{22}*B_{2}+a_{23}*B_{3} │ └ a_{31}*B_{1}+a_{32}*B_{2}+a_{33}*B_{3} ┘ |
= | [ C_{1} C_{2} C_{3} ] |
|B > < A| | = |
┌ B_{1} ┐ │ B_{2} │ └ B_{3} ┘ |
[ A_{1} A_{2} A_{3} ] | = |
┌ B_{1}*A_{1}+B_{1}*A_{2}+B_{1}*A_{3} ┐ │ B_{2}*A_{1}+B_{2}*A_{2}+B_{2}*A_{3} │ └ B_{3}*A_{1}+B_{3}*A_{2}+B_{3}*A_{3} ┘ |