Coordinatis orthogonalibus et valoribus realibus
His vectoribus iuxta basem orthogonalem scriptis
,
productum scribi potest
![{\displaystyle \langle {\vec {a)),{\vec {b))\rangle ={\vec {a))^{T}\,{\vec {b))={\begin{bmatrix}a_{1}\,a_{2}\,a_{3}\,\dots \end{bmatrix)){\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\\vdots \end{bmatrix))=\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n))](https://wikimedia.org/api/rest_v1/media/math/render/svg/5defed4c01b021d47cc3e2403d014664c7e8eb8e)
ubi T denotat transpositionem matricis, Σ denotat summam arithmeticam et n est dimensio spatii vectorialis.
Coordinatis orthogonalibus et valoribus complexis
His autem vectoribus valoribus complexis praeditis, productum interius scribi oportet
![{\displaystyle \langle {\vec {a)),{\vec {b))\rangle ={\vec {a))^{\dagger }\,{\vec {b))={\begin{bmatrix}a_{1}^{*}\,a_{2}^{*}\,a_{3}^{*}\,\dots \end{bmatrix)){\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\\vdots \end{bmatrix))=\sum _{i=1}^{n}a_{i}^{*}b_{i}=a_{1}^{*}b_{1}+a_{2}^{*}b_{2}+\cdots +a_{n}^{*}b_{n))](https://wikimedia.org/api/rest_v1/media/math/render/svg/b207cbd6b55d602741a18c5539828d85f57d9771)
ubi * denotat coniugationem complexam et † denotat simultaneam coniugationem et transpositionem. Hac definitione maxime numeris complexis accomodata effecit ut semper scribi possit valore scalari reali
![{\displaystyle {\vec {a))\cdot {\vec {a))=\left\|{\vec {a))\right\|^{2))](https://wikimedia.org/api/rest_v1/media/math/render/svg/18b805ad671bdbc04c4ac816929d392c2fe55bf2)