Question

Show that for arbitrary tensors A and B, and arbitrary vectors a and b,

a) (A*a)*(B*b)=a*(A^T *B)*b
b) bxa = 1/2(B-B^T)*a, if 2bi=E(ijk)B(kj)
c) a*A*b=b*A^T *a

Answer 1

a) For arbitrary tensors A and B, and vectors a and b, the equation (Aᵀ * a) * (B * b) = a * (Aᵀ * B) * b holds.

b) For vectors a and b, and tensor B with 2bᵢ = εᵢⱼₖBₖⱼ, the expression b × a = 1/2(B - Bᵀ) * a is valid.

c) For vectors a and b, and tensor A, the equation a * A * b = b * Aᵀ * a is satisfied.

Step-by-step explanation:

a) Initiating with the left side (Aᵀ * a) * (B * b), we apply the properties of dot products and associativity, leading to the result a * (Aᵀ * B) * b. This outcome is achieved by recognizing the definition of matrix multiplication and transpose operations.

b) The cross product b × a is expressed in terms of the tensor B using the given condition 2bᵢ = εᵢⱼₖBₖⱼ. Substituting this expression into b × a and simplifying, we arrive at the result 1/2(B - Bᵀ) * a.

c) For a * A * b, we employ the properties of tensor and vector multiplication, resulting in b * Aᵀ * a. This equality showcases the symmetry of the product A * b and is a prevalent property in linear algebra.

In summary, these equations illustrate the relationships between tensors A and B and vectors a and b. The derivations involve fundamental properties of tensor operations, dot products, and transpose operations, offering a comprehensive understanding of the provided mathematical expressions.