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Generalized Product of Arrays—Wolfram Documentation

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• See Also
  • Dot
  • TensorContract
  • Inner
  • D
  • ArraySymbol
  • SymbolicIdentityArray
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  • Symbolic Vectors, Matrices and Arrays

ArrayDot[a,b,k]

computes the product of arrays a and b obtained by summing up products of terms over the last k dimensions of a and the first k dimensions of b.

ArrayDot[a,b,{{s₁,t₁},{s₂,t₂},…}]

computes the product of arrays a and b obtained by summing up products of terms over the pairs {s_i,t_i} of dimensions.

• The arguments a and b in ArrayDot[a,b,k] should be arrays with dimensions, respectively, {m₁,…,m_p,d₁,…,d_k} and {d₁,…,d_k,n₁,…,n_q}. The result is an array c with dimensions {m₁,…,m_p,n₁,…,n_q} and c_{i1,…,ip,j1,…,jq}a_{i1,…,ip,α1,…,αk}b_{α1,…,αk,j1,…,jq}.
• In ArrayDot[a,b,{{s₁,t₁},…,{s_k,t_k}}], the arguments a and b should be arrays with dimensions, respectively, {m₁,…,m_p} and {n₁,…,n_q}, where all s_i are distinct, all t_i are distinct and for all i, 1≤s_i≤p, 1≤t_i≤q and m_sin_ti. The result is equal to TensorContract[ab,{{s₁,p+t₁},…,{s_k,p+t_k}}].
• ArrayDot can be used on SparseArray and structured array objects.
• ArrayDot is used in the array differentiation chain rule.

Compute ArrayDot over two dimensions:

Compute ArrayDot over specified pairs of dimensions:

ArrayDot is used in the array differentiation chain rule:

Exact number arrays:

Real machine-number arrays:

Complex arrays:

Symbolic arrays:

Finite field element arrays:

CenteredInterval arrays:

Find random representatives arep and brep of a and b:

Verify that ArrayDot[a,b,3] contains ArrayDot[arep,brep,3]:

ArrayDot of sparse arrays is another sparse array:

Format the result:

Structured arrays:

Efficiently multiply large arrays:

Approximate the determinant of a perturbed matrix:

Order zero approximation:

Compare with the exact value:

Order one approximation:

Compare with the exact value:

Order two approximation:

Compare with the exact value:

ArrayDot is linear in each argument:

Dot[a,b] is equal to ArrayDot[a,b,1]:

SymbolicIdentityArray objects are identity elements for ArrayDot:

For a real matrix a, Norm[a,"Frobenius"] is equal to the square root of ArrayDot[a,a,2]:

If c=ArrayDot[a,b, k], then c_{i1,…,ip,j1,…,jq}a_{i1,…,ip,α1,…,αk}b_{α1,…,αk,j1,…,jq}:

ArrayDepth[ArrayDot[a,b,k]] is equal to ArrayDepth[a]+ArrayDepth[b]-2k:

ArrayDot can be implemented as a combination of TensorProduct and TensorContract:

ArrayDot can be implemented as a combination of Flatten and Dot:

ArrayDot is used in the array differentiation chain rule:

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