Kronecker Product: Outer Product of Matrices

1. Kronecker Product or Outer Product of Matrices can be carried out between Any 2 Matrices of Any Dimensions.
2. Let's consider 2 Column Matrices (or Vectors), a \(M \times 1\) Matrix \(A\) and a \(N \times 1\) Matrix \(B\) given as follows
   
   \(A=\begin{bmatrix}a_{1} \\ a_{2} \\ \vdots \\ a_m\end{bmatrix}\hspace{.5cm}B=\begin{bmatrix}b_{1} \\ b_{2} \\ \vdots \\ b_n\end{bmatrix}\)
   
   The Kronecker Product \(C=A \otimes B\) is calculated by Multiplying Each Element of Vector \(A\) with All the Elements of Vector \(B\) as given below
   
   \(C=A \otimes B =\begin{bmatrix}a_{1} \\ a_{2} \\ \vdots \\ a_m\end{bmatrix} \otimes \begin{bmatrix}b_{1} \\ b_{2} \\ \vdots \\ b_n\end{bmatrix}=\begin{bmatrix}a_1 \cdot b_1 \\ a_1 \cdot b_2 \\ \vdots \\ a_1 \cdot b_n \\ a_2 \cdot b_1 \\ a_2 \cdot b_2 \\ \vdots \\ a_2 \cdot b_n \\ \vdots \\ a_m \cdot b_1 \\ a_m \cdot b_2 \\ \vdots \\ a_m \cdot b_n\end{bmatrix}\)
   
   Similarly, The Kronecker Product \(D=B \otimes A\) is calculated by Multiplying Each Element of Vector \(B\) with All the Elements of Vector \(A\) as given below
   
   \(D=B \otimes A =\begin{bmatrix}b_{1} \\ b_{2} \\ \vdots \\ b_n\end{bmatrix}\otimes \begin{bmatrix}a_{1} \\ a_{2} \\ \vdots \\ a_m\end{bmatrix} =\begin{bmatrix}b_1 \cdot a_1 \\ b_1 \cdot a_2 \\ \vdots \\ b_1 \cdot a_m \\ b_2 \cdot a_1 \\ b_2 \cdot a_2 \\ \vdots \\ b_2 \cdot a_m \\ \vdots \\ b_n \cdot a_1 \\ b_n \cdot a_2 \\ \vdots \\ b_n \cdot a_m\end{bmatrix}\)
   
   The Resultant Column Matrices \(C\) and \(D\) Both have the Dimensions of \(MN \times 1\).
3. Let's consider 2 Row Matrices (or Co-Vectors), a \(1 \times M\) Matrix \(A\) and a \(1 \times N\) Matrix \(B\) given as follows
   
   \(A=\begin{bmatrix}a_{1} & a_{2} & \cdots & a_m\end{bmatrix}\hspace{.5cm}B=\begin{bmatrix}b_{1} & b_{2} & \cdots & b_n\end{bmatrix}\)
   
   The Kronecker Product \(C=A \otimes B\) is calculated by Multiplying Each Element of Co-Vector \(A\) with All the Elements of Co-Vector \(B\) as given below
   
   \(C=A \otimes B =\begin{bmatrix}a_{1} & a_{2} & \cdots & a_m\end{bmatrix} \otimes \begin{bmatrix}b_{1} & b_{2} & \cdots & b_n\end{bmatrix}=\begin{bmatrix}a_1 \cdot b_1 & a_1 \cdot b_2 & \cdots & a_1 \cdot b_n & a_2 \cdot b_1 & a_2 \cdot b_2 & \cdots & a_2 \cdot b_n & \cdots & a_m \cdot b_1 & a_m \cdot b_2 & \cdots & a_m \cdot b_n\end{bmatrix}\)
   
   Similarly, The Kronecker Product \(D=B \otimes A\) is calculated by Multiplying Each Element of Co-Vector \(B\) with All the Elements of Co-Vector \(A\) as given below
   
   \(D=B \otimes A =\begin{bmatrix}b_{1} & b_{2} & \cdots & b_n\end{bmatrix}\otimes \begin{bmatrix}a_{1} & a_{2} & \cdots & a_m\end{bmatrix} =\begin{bmatrix}b_1 \cdot a_1 & b_1 \cdot a_2 & \cdots & b_1 \cdot a_m & b_2 \cdot a_1 & b_2 \cdot a_2 & \cdots & b_2 \cdot a_m & \cdots & b_n \cdot a_1 & b_n \cdot a_2 & \cdots & b_n \cdot a_m\end{bmatrix}\)
   
   The Resultant Row Matrices \(C\) and \(D\) Both have the Dimensions of \(1 \times MN\).
4. Let's consider 2 Matrices, a \(M \times N\) Matrix \(A\) containing elements \(a_{ij}\) and a \(P \times Q\) Matrix \(B\) containing elements \(b_{ij}\) given as follows
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ a_{31} & a_{32} & ... & a_{3n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \hspace{.5cm}B=\begin{bmatrix} b_{11} & b_{12} & ... & b_{1q}\\ b_{21} & b_{22} & ... & b_{2q} \\ b_{31} & b_{32} & ... & b_{3q} \\ \vdots & \vdots & \ddots & \vdots \\b_{p1} & b_{p2} & ... & b_{pq}\end{bmatrix}\)
   
   The Kronecker Product \(C=A \otimes B\) is calculated by Multiplying Each Element of Matrix \(A\) with All the Elements of Matrix \(B\) given as follows
   
   \(C=A \otimes B=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ a_{31} & a_{32} & ... & a_{3n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \otimes \begin{bmatrix} b_{11} & b_{12} & ... & b_{1q}\\ b_{21} & b_{22} & ... & b_{2q} \\ b_{31} & b_{32} & ... & b_{3q} \\ \vdots & \vdots & \ddots & \vdots \\b_{p1} & b_{p2} & ... & b_{pq}\end{bmatrix}= \begin{bmatrix} a_{11}\textbf{B} & a_{12}\textbf{B} & ... & a_{1n}\textbf{B}\\ a_{21}\textbf{B} & a_{22}\textbf{B} & ... & a_{2n}\textbf{B} \\ a_{31}\textbf{B} & a_{32}\textbf{B} & ... & a_{3n}\textbf{B} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1}\textbf{B} & a_{m2}\textbf{B} & ... & a_{mn}\textbf{B}\end{bmatrix} \)
   
   The Kronecker Product \(D=B \otimes A\) is calculated is calculated by Multiplying Each Element of Matrix \(B\) with All the Elements of Matrix \(A\) given as follows
   
   \(D=B \otimes A=\begin{bmatrix} b_{11} & b_{12} & ... & b_{1q}\\ b_{21} & b_{22} & ... & b_{2q} \\ b_{31} & b_{32} & ... & b_{3q} \\ \vdots & \vdots & \ddots & \vdots \\b_{p1} & b_{p2} & ... & b_{pq}\end{bmatrix} \otimes \begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ a_{31} & a_{32} & ... & a_{3n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix}= \begin{bmatrix} b_{11}\textbf{A} & b_{12}\textbf{A} & ... & b_{1q}\textbf{A}\\ b_{21}\textbf{A} & b_{22}\textbf{A} & ... & b_{2q}\textbf{A} \\ b_{31}\textbf{A} & b_{32}\textbf{A} & ... & b_{3q}\textbf{A} \\ \vdots & \vdots & \ddots & \vdots \\b_{p1}\textbf{A} & b_{p2}\textbf{A} & ... & b_{pq}\textbf{A}\end{bmatrix} \)
   
   The Resultant Matrices \(C\) and \(D\) Both have the Dimensions of \(MP \times NQ\)
   
   An alternate way to calculate the the Kronecker Product is to consider Matrix \(A\) to contain Vectors \(\textbf{a}_1, \textbf{a}_2 ,..., \textbf{a}_m\) and Matrix \(B\) to contain Vectors \(\textbf{b}_1, \textbf{b}_2 ,..., \textbf{b}_n\). Then the Kronecker Product \(C=A \otimes B\) is calculated as following
   
   \(C=A \otimes B = \begin{bmatrix} \textbf{a}_1 \otimes \textbf{b}_1 & \textbf{a}_1 \otimes \textbf{b}_2 & \cdots & \textbf{a}_1 \otimes \textbf{b}_n & \textbf{a}_2 \otimes \textbf{b}_1 & \textbf{a}_2 \otimes \textbf{b}_2 & \cdots & \textbf{a}_2 \otimes \textbf{b}_n & \cdots & \textbf{a}_m \otimes \textbf{b}_1 & \textbf{a}_m \otimes \textbf{b}_2 \cdots & \textbf{a}_m \otimes \textbf{b}_n\end{bmatrix}\)
   
   Similarly, the Kronecker Product \(D=B \otimes A\) is calculated as following
   
   \(D=B \otimes A = \begin{bmatrix} \textbf{b}_1 \otimes \textbf{a}_1 & \textbf{b}_1 \otimes \textbf{a}_2 & \cdots & \textbf{b}_1 \otimes \textbf{a}_m & \textbf{b}_2 \otimes \textbf{a}_1 & \textbf{b}_2 \otimes \textbf{a}_2 & \cdots & \textbf{b}_2 \otimes \textbf{a}_m & \cdots & \textbf{b}_n \otimes \textbf{a}_1 & \textbf{b}_n \otimes \textbf{a}_2 \cdots & \textbf{b}_n \otimes \textbf{a}_m\end{bmatrix}\)
   
   Another alternate way to calculate the the Kronecker Product is to consider Matrix \(A\) to contain Co-Vectors \(\textbf{a}_1, \textbf{a}_2 ,..., \textbf{a}_m\) and Matrix \(B\) to contain Co-Vectors \(\textbf{b}_1, \textbf{b}_2 ,..., \textbf{b}_n\). Then the Kronecker Product \(C=A \otimes B\) is calculated as following
   
   \(C=A \otimes B = \begin{bmatrix} \textbf{a}_1 \otimes \textbf{b}_1 \\ \textbf{a}_1 \otimes \textbf{b}_2 \\ \vdots \\ \textbf{a}_1 \otimes \textbf{b}_n \\ \textbf{a}_2 \otimes \textbf{b}_1 \\ \textbf{a}_2 \otimes \textbf{b}_2 \\ \vdots \\ \textbf{a}_2 \otimes \textbf{b}_n \\ \vdots \\ \textbf{a}_m \otimes \textbf{b}_1 \\ \textbf{a}_m \otimes \textbf{b}_2 \\ \vdots \\ \textbf{a}_m \otimes \textbf{b}_n\end{bmatrix}\)
   
   Similarly, the Kronecker Product \(D=B \otimes A\) is calculated as following
   
   \(D=B \otimes A = \begin{bmatrix} \textbf{b}_1 \otimes \textbf{a}_1 \\ \textbf{b}_1 \otimes \textbf{a}_2 \\ \vdots \\ \textbf{b}_1 \otimes \textbf{a}_m \\ \textbf{b}_2 \otimes \textbf{a}_1 \\ \textbf{b}_2 \otimes \textbf{a}_2 \\ \vdots \\ \textbf{b}_2 \otimes \textbf{a}_m \\ \vdots \\ \textbf{b}_n \otimes \textbf{a}_1 \\ \textbf{b}_n \otimes \textbf{a}_2 \\ \vdots \\ \textbf{b}_n \otimes \textbf{a}_m\end{bmatrix}\)
5. Following example calculates the Kronecker Products of 2 Matrices \(A\) and \(B\) as given below
   
   \(A=\begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix}\hspace{.5cm}B=\begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix}\)
   
   \(A \otimes B=\begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix} \otimes \begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix}\)
   
   \(\Rightarrow A \otimes B=\begin{bmatrix} -2 \begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix} & 3\begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix} & 7 \begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix}\\3\begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix} & -5 \begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix} & 9 \begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix}\end{bmatrix}\)
   
   \(\Rightarrow A \otimes B=\begin{bmatrix} \begin{bmatrix}-2\\3\end{bmatrix} \otimes \begin{bmatrix}5\\1\\8\end{bmatrix} & \begin{bmatrix}-2\\3\end{bmatrix} \otimes \begin{bmatrix}-3\\6\\3\end{bmatrix} & \begin{bmatrix}3\\-5\end{bmatrix} \otimes \begin{bmatrix}5\\1\\8\end{bmatrix} & \begin{bmatrix}3\\-5\end{bmatrix} \otimes \begin{bmatrix}-3\\6\\3\end{bmatrix} & \begin{bmatrix}7\\9\end{bmatrix} \otimes \begin{bmatrix}5\\1\\8\end{bmatrix} & \begin{bmatrix}7\\9\end{bmatrix} \otimes \begin{bmatrix}-3\\6\\3\end{bmatrix} \end{bmatrix}\)
   
   \(\Rightarrow A \otimes B=\begin{bmatrix} \begin{bmatrix}-2 & 3 & 7\end{bmatrix} \otimes \begin{bmatrix}5 & -3\end{bmatrix} \\ \begin{bmatrix}-2 & 3 & 7\end{bmatrix} \otimes \begin{bmatrix}1 & 6\end{bmatrix} \\ \begin{bmatrix}-2 & 3 & 7\end{bmatrix} \otimes \begin{bmatrix}8 & 3\end{bmatrix} \\ \begin{bmatrix}3 & -5 & 9\end{bmatrix} \otimes \begin{bmatrix}5 & -3\end{bmatrix} \\ \begin{bmatrix}3 & -5 & 9\end{bmatrix} \otimes \begin{bmatrix}1 & 6\end{bmatrix} \\ \begin{bmatrix}3 & -5 & 9\end{bmatrix} \otimes \begin{bmatrix}8 & 3\end{bmatrix} \end{bmatrix}\)
   
   \(\Rightarrow A \otimes B=\begin{bmatrix}-10 & 6 & 15 & -9 & 35 & -21 \\ -2 & -12 & 3 & 18 & 7 & 42 \\ -16 & -6 & 24 & 9 & 56 & 21 \\ 15 & -9 & -25 & 15 & 45 & -27 \\ 3 & 18 & -5 & -30 & 9 & 54 \\ 24 & 9 & -40 & -15 & 72 & 27\end{bmatrix}\)
   
   \(B \otimes A=\begin{bmatrix}5 & -3 \\1 & 6 \\ 8 & 3\end{bmatrix} \otimes \begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix}\)
   
   \(\Rightarrow B \otimes A=\begin{bmatrix} 5 \begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix} & -3 \begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix} \\ 1 \begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix} & 6 \begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix} \\ 8 \begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix} & 3 \begin{bmatrix} -2 & 3 & 7\\3 & -5 & 9\end{bmatrix} \end{bmatrix}\)
   
   \(\Rightarrow B \otimes A=\begin{bmatrix} \begin{bmatrix}5\\1\\8\end{bmatrix} \otimes \begin{bmatrix}-2\\3\end{bmatrix} & \begin{bmatrix}5\\1\\8\end{bmatrix} \otimes \begin{bmatrix}3\\-5\end{bmatrix} & \begin{bmatrix}5\\1\\8\end{bmatrix} \otimes \begin{bmatrix}7\\9\end{bmatrix} & \begin{bmatrix}-3\\6\\3\end{bmatrix} \otimes \begin{bmatrix}-2\\3\end{bmatrix} & \begin{bmatrix}-3\\6\\3\end{bmatrix} \otimes \begin{bmatrix}3\\-5\end{bmatrix} & \begin{bmatrix}-3\\6\\3\end{bmatrix} \otimes \begin{bmatrix}7\\9\end{bmatrix} \end{bmatrix}\)
   
   \(\Rightarrow A \otimes B=\begin{bmatrix} \begin{bmatrix}5 & -3\end{bmatrix} \otimes \begin{bmatrix}-2 & 3 & 7\end{bmatrix} \\ \begin{bmatrix}5 & -3\end{bmatrix} \otimes \begin{bmatrix}3 & -5 & 9\end{bmatrix} \\ \begin{bmatrix}1 & 6\end{bmatrix} \otimes \begin{bmatrix}-2 & 3 & 7\end{bmatrix} \\ \begin{bmatrix}1 & 6\end{bmatrix} \otimes \begin{bmatrix}3 & -5 & 9\end{bmatrix} \\ \begin{bmatrix}8 & 3\end{bmatrix}\otimes \begin{bmatrix}-2 & 3 & 7\end{bmatrix} \\ \begin{bmatrix}8 & 3\end{bmatrix} \otimes \begin{bmatrix}3 & -5 & 9\end{bmatrix} \end{bmatrix}\)
   
   \(\Rightarrow B \otimes A=\begin{bmatrix}-10 & 15 & 35 & 6 & -9 & -21 \\ 15 & -25 & 45 & -9 & 15 & -27 \\ -2 & 3 & 7 & -12 & 18 & 42 \\ 3 & -5 & 9 & 18 & -30 & 54 \\ -16 & 24 & 56 & -6 & 9 & 21 \\ 24 & -40 & 72 & 9 & -15 & 27\end{bmatrix}\)
6. You can use the Matrix Kronecker Product / Outer Product Calculator to calculate Kronecker Product / Outer Product of Matrices.