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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2015 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.