Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a special case of a tensor product. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.

Definition

If A is an m-by-n matrix and B is a p-by-q matrix, then the Kronecker product A ⊗ B is the mp-by-nq block matrix

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: A otimes B = egin{bmatrix} a_{11} B & cdots & a_{1n}B \ dots & ddots & dots \ a_{m1} B & cdots & a_{mn} B end{bmatrix}.

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More explicitly, we have

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: A otimes B = egin{bmatrix}

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a_{11} b_{11} & a_{11} b_{12} & cdots & a_{11} b_{1q} &

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cdots & cdots & a_{1n} b_{11} & a_{1n} b_{12} & cdots & a_{1n} b_{1q} \

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a_{11} b_{21} & a_{11} b_{22} & cdots & a_{11} b_{2q} &

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cdots & cdots & a_{1n} b_{21} & a_{1n} b_{22} & cdots & a_{1n} b_{2q} \

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dots & dots & ddots & dots & & & dots & dots & ddots & dots \

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a_{11} b_{p1} & a_{11} b_{p2} & cdots & a_{11} b_{pq} &

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cdots & cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & cdots & a_{1n} b_{pq} \

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dots & dots & & dots & ddots & & dots & dots & & dots \

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dots & dots & & dots & & ddots & dots & dots & & dots \

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a_{m1} b_{11} & a_{m1} b_{12} & cdots & a_{m1} b_{1q} &

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cdots & cdots & a_{mn} b_{11} & a_{mn} b_{12} & cdots & a_{mn} b_{1q} \

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a_{m1} b_{21} & a_{m1} b_{22} & cdots & a_{m1} b_{2q} &

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cdots & cdots & a_{mn} b_{21} & a_{mn} b_{22} & cdots & a_{mn} b_{2q} \

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dots & dots & ddots & dots & & & dots & dots & ddots & dots \

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a_{m1} b_{p1} & a_{m1} b_{p2} & cdots & a_{m1} b_{pq} &

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cdots & cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & cdots & a_{mn} b_{pq}

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end{bmatrix}.

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Examples

:

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egin{bmatrix}

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1 & 2 \

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3 & 1 \

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end{bmatrix}

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otimes

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egin{bmatrix}

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0 & 3 \

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2 & 1 \

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end{bmatrix}

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