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In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.
The Kronecker product is named after
so that:
A column-wise Kronecker product of two matrices may also be called the Khatri–Rao product. This product assumes the partitions of the matrices are their columns. In this case m_{1} = m, p_{1} = p, n = q and for each j: n_{j} = p_{j} = 1. The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B. Using the matrices from the previous examples with the columns partitioned:
This is a submatrix of the Tracy–Singh product of the two matrices (each partition in this example is a partition in a corner of the Tracy–Singh product).
in which the ij-th block is the m_{i}p_{i} × n_{j}q_{j} sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal. The size of the product is then (Σ_{i} m_{i}p_{i}) × (Σ_{j} n_{j}q_{j}). Proceeding with the same matrices as the previous example we obtain:
The Khatri–Rao product is defined as^{[8]}^{[9]}
we get:
For example, if A and B both are 2 × 2 partitioned matrices e.g.:
which means that the (ij)-th subblock of the mp × nq product A ∘ B is the m_{i} p × n_{j} q matrix A_{ij} ∘ B, of which the (kℓ)-th subblock equals the m_{i} p_{k} × n_{j} q_{ℓ} matrix A_{ij} ⊗ B_{kℓ}. Essentially the Tracy–Singh product is the pairwise Kronecker product for each pair of partitions in the two matrices.
The Tracy–Singh product is defined as^{[6]}^{[7]}
Two related matrix operations are the Tracy–Singh and Khatri–Rao products which operate on partitioned matrices. Let the m × n matrix A be partitioned into the m_{i} × n_{j} blocks A_{ij} and p × q matrix B into the p_{k} × q_{ℓ} blocks B_{kl} with of course Σ_{i} m_{i} = m, Σ_{j} n_{j} = n, Σ_{k} p_{k} = p and Σ_{ℓ} q_{ℓ} = q.
For an example of the application of this formula, see the article on the Lyapunov equation. This formula also comes in handy in showing that the matrix normal distribution is a special case of the multivariate normal distribution.
If X is row-ordered into the column vector x then AXB can be also be written as (Jain 1989, 2.8 Block Matrices and Kronecker Products) (A ⊗ B^{T})x.
Here, vec(X) denotes the vectorization of the matrix X formed by stacking the columns of X into a single column vector. It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution if and only if A and B are nonsingular (Horn & Johnson 1991, Lemma 4.3.1).
The Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation AXB = C, where A, B and C are given matrices and the matrix X is the unknown. We can rewrite this equation as
It follows that the trace and determinant of a Kronecker product are given by
Similarly, denote the nonzero singular values of B by
Then the Kronecker product A ⊗ B has r_{A}r_{B} nonzero singular values, namely
Since the rank of a matrix equals the number of nonzero singular values, we find that
This is called the mixed-product property, because it mixes the ordinary matrix product and the Kronecker product. It follows that A ⊗ B is invertible if and only if both A and B are invertible, in which case the inverse is given by
Note that this is different from the direct sum of two matrices. This operation is related to the tensor product on Lie algebras.
We have the following formula for the matrix exponential, which is useful in some numerical evaluations.^{[3]}
Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let H^{i} be the Hamiltonian of the ith such system. Then the total Hamiltonian of the ensemble is
If A and B represent linear transformations V_{1} → W_{1} and V_{2} → W_{2}, respectively, then A ⊗ B represents the tensor product of the two maps, V_{1} ⊗ V_{2} → W_{1} ⊗ W_{2}.
More compactly, we have (A\otimes B)_{p(r-1)+v, q(s-1)+w} = a_{rs} b_{vw}
more explicitly:
If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the mp × nq block matrix:
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