1.12. Tensor Products 13 1.12. Tensor Products More conceptual and abstract approaches to tensor products may be found else- where. For our purposes, a coordinate approach will suﬃce. Let V and W be finite-dimensional vector spaces over a field F. Let m = dim V and n = dim V. Then we may consider V to be Fm and W to be Fn. For any ordered pair of column vectors (v, w), where v ∈ V and w ∈ W, we define the tensor product v ⊗ w to be the column vector ⎛ ⎜ ⎜ ⎝ w1v w2v . . wnv ⎞ ⎟ ⎟ ⎠ . This vector has mn coordinates. The tensor product of V and W is the space spanned by all possible vectors v ⊗ w and is denoted V ⊗ W. Letting v and w run through the unit coordinate vectors in V and W, respectively, it is clear that V ⊗W contains the mn unit coordinate vectors of Fmn, and so V ⊗ W is mn-dimensional space over F. Let S : V → V and T : W → W be linear transformations. We define (S ⊗ T )(v ⊗ w) = (S(v)) ⊗ (T (w)) and extend by linearity to define a linear transformation S ⊗ T on V ⊗ W. Let A be an m × n matrix, and let B be an r × s matrix. The tensor product of A and B is the mr × ns matrix (1.13) A ⊗ B = ⎛ ⎜ ⎜ ⎝ b11A b12A · · · b1sA b21A b22A · · · b2sA . . . . · · · . . br1A br2A · · · brsA ⎞ ⎟ ⎟ ⎠ . In (1.13), the (i, j) block is Abij there are rs blocks, each of size m × n. For example, a b c d ⊗ 1 2 3 4 5 6 = ⎛ ⎜ ⎝ a b 2a 2b 3a 3b c d 2c 2d 3c 3d 4a 4b 5a 5b 6a 6b 4c 4d 5c 5d 6c 6d ⎞ ⎟ ⎠ . If A and B are square matrices of sizes m×m and r ×r, respectively, this definition of A ⊗ B will give (A ⊗ B)(v ⊗ w) = (Av) ⊗ (Bw) for coordinate vectors v ∈ Fm and w ∈ Fr. More generally, for matrices A, B, C, D, of sizes m × n, r × s, n × p, and s × q, respectively, we have the formula (1.14) (A ⊗ B)(C ⊗ D) = (AC) × (BD). This can be established by direct computation using block multiplication.

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