Wikipedia, Entziklopedia askea
Aljebra linealean, matrize karratu baten aztarna bere diagonal nagusiko elementuen batura da.
Hau da,
![{\displaystyle \operatorname {tr} (A)=a_{11}+a_{22}+\dots +a_{nn}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d4f5afb5d0ae849e655e2885168bb8da583092)
non aij i-garren errenkadan eta j-garren zutabean dagoen elementua den.
![{\displaystyle A={\begin{pmatrix}3&0&1\\0&4&5\\0&1&2\end{pmatrix}}\Rightarrow {\mbox{tr}}(A)=3+4+2=9.\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e49c98bb12ae420a1c520f82f794bac4d2ef5b1)
![{\displaystyle B={\begin{pmatrix}3&0&1\\0&4&5\\0&1&-7\end{pmatrix}}\Rightarrow {\mbox{tr}}(B)=3+4+(-7)=0.\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65e3f9e27edd86d380898f52ebdbfca86c593846)
![{\displaystyle \operatorname {tr} \left(A+B\right)=\operatorname {tr} \left(A\right)+\operatorname {tr} \left(B\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddc723c9db1d17f4c408920e56832ae95cf8d89)
![{\displaystyle \operatorname {tr} \left(rA\right)=r\left(\operatorname {tr} \left(A\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c19934da18914bbb3ebfc50d54ad69c8431850f7)
eta
matrize karratuak izanik, eta
eskalar bat.
- Matrizea iraultzeak ez duenez eraginik sortzen diagonal nagusian,
![{\displaystyle \operatorname {tr} \left(A^{T}\right)=\operatorname {tr} \left(A\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5350a0b8d70503bca57621ab76ec207f461b393)
dimentsioko matrize bat bada eta
dimentsiokoa, orduan
![{\displaystyle \operatorname {tr} \left(AB\right)=\operatorname {tr} \left(BA\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb7e9ac4c66c2d55f12936ce82242623ea7ece9)
- Frogatzeko, aintzat hartu behar dugu
eta
matrizeen biderkadura honelakoa dela
![{\displaystyle [AB]_{ij}=\sum _{k=1}^{m}[A]_{ik}[B]_{kj}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786fb5609562fee995fcc37c8f16e5034c6658cc)
- hortaz,
-ren aztarna honela adieraz dezakegu
![{\displaystyle \operatorname {tr} \left(AB\right)=\sum _{i=1}^{n}[AB]_{ii}=\sum _{i=1}^{n}\sum _{k=1}^{m}[A]_{ik}[B]_{ki}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ed25dfd68c552d1fb90d199d067db80b6b4283)
- eta batuketaren elkartze-propietatea kontuan hartuz gero
![{\displaystyle \operatorname {tr} \left(AB\right)=\sum _{k=1}^{m}\sum _{i=1}^{n}[A]_{ik}[B]_{ki}=\sum _{k=1}^{m}\sum _{i=1}^{n}[B]_{ki}[A]_{ik}=\sum _{k=1}^{m}[AB]_{kk}=\operatorname {tr} \left(BA\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0f234020d2888703d23c253d7f22e0065a3fa2)
- Azpimarratu behar dugu
dimentsioko matrize karratu bat dela, eta
, aldiz,
dimentsioko matrize bat.