vector direct product: increasing rank
vector direct product = rank-1 tensor times rank-1 tensor equals rank-2 tensor: increasing rank
U⊗V=UV⊺=UiVj=⎛⎜⎝U1U2U3⎞⎟⎠⊗⎛⎜⎝V1V2V3⎞⎟⎠=⎛⎜⎝U1U2U3⎞⎟⎠(V1V2V3)=⎛⎜⎝U1V1U1V2U1V3U2V1U2V2U2V3U3V1U3V2U3V3⎞⎟⎠=⎛⎜⎝U1V⊺U2V⊺U3V⊺⎞⎟⎠=(UV1UV2UV3)
vector inner product: decreasing rank
vector inner product = rank-1 tensor times rank-1 tensor equals rank-0 tensor: decreasing rank
U⋅V=V⊺U=ViUi=⎛⎜⎝U1U2U3⎞⎟⎠⋅⎛⎜⎝V1V2V3⎞⎟⎠=(V1V2V3)⎛⎜⎝U1U2U3⎞⎟⎠=V1U1+V2U2+V3U3
tensor direct product: increasing rank
S⊗T=SikTjl(i,j),(k,l)∈{(1,1),(1,2),(2,1),(2,2)}(S11S12S21S22)⊗(T11T12T21T22)=⎛⎜
⎜
⎜⎝S11T11S11T12S12T11S12T12S11T21S11T22S12T21S12T22S21T11S21T12S22T11S22T12S21T21S21T22S22T21S22T22⎞⎟
⎟
⎟⎠(S11S12S21S22)(T11T12T21T22)=(S11TS12TS21TS22T)=⎛⎜
⎜
⎜⎝S11(T11T12T21T22)S12(T11T12T21T22)S21(T11T12T21T22)S22(T11T12T21T22)⎞⎟
⎟
⎟⎠