19th Juin2022

by carodah

## AutoCAD Full Version For Windows

Q: How do I take the kronecker product of a vector and a matrix? How can I take the Kronecker product of a vector and a matrix? A: The Kronecker product of a matrix with a vector is a vector. If $C = C_1 \otimes C_2$ is the Kronecker product of two matrices, then $C$ is a square matrix. The $k$-th row of $C$ is a vector $v_k = C_1 \otimes e_k$, where $\{e_k\}_{k=1}^n$ are the standard basis vectors. The $l$-th column of $C$ is a vector $u_l = e_l \otimes C_2$, where $\{e_k\}_{k=1}^n$ are the standard basis vectors. Notice that $$u_l(k) = \sum_{i=1}^n C_2(i,l) e_i(k)$$ That is, $C$ is the matrix of the linear map from $\Bbb{R}^n$ to $\Bbb{R}^n$ with $C_1$ as the input and $C_2$ as the output. EDIT: I have just realized that the OP is asking for a proof that $C$ is a matrix. The answer is yes, but we need a little more than what I wrote above. We need to show that $C_1 \otimes C_2$ is a linear map from $\Bbb{R}^n \times \Bbb{R}^n$ to $\Bbb{R}^n \times \Bbb{R}^n$. Recall that the linear map $\varphi : \Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R}^n \times \Bbb{R}^n$ such that $\varphi(x,y) = (Ax + By, Cx + Dy)$ for some real $n$-by-$n$ matrices $A,B,C,D$. We can write the Kronecker product as  \varphi(x,y) =