Category:Definitions/Matrix Products

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This category contains definitions related to Matrix Products.
Related results can be found in Category:Matrix Products.


Matrix Product (Conventional)

Let $\struct {R, +, \circ}$ be a ring.

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$.

Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$.

Then the matrix product of $\mathbf A$ and $\mathbf B$ is written $\mathbf A \mathbf B$ and is defined as follows.

Let $\mathbf A \mathbf B = \mathbf C = \sqbrk c_{m p}$.


Then:

$\ds \forall i \in \closedint 1 m, j \in \closedint 1 p: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$


Thus $\sqbrk c_{m p}$ is the $m \times p$ matrix where each entry $c_{i j}$ is built by forming the (ring) product of each entry in the $i$'th row of $\mathbf A$ with the corresponding entry in the $j$'th column of $\mathbf B$ and adding up all those products.


This operation is called matrix multiplication, and $\mathbf C$ is the matrix product of $\mathbf A$ with $\mathbf B$.


Matrix Scalar Product

Let $\GF$ denote one of the standard number systems.

Let $\map \MM {m, n}$ be the $m \times n$ matrix space over $\GF$.

Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.

Let $\lambda \in \GF$ be any element of $\Bbb F$.


The operation of scalar multiplication of $\mathbf A$ by $\lambda$ is defined as follows.

Let $\lambda \mathbf A = \mathbf C$.

Then:

$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = \lambda a_{i j}$

$\lambda \mathbf A$ is the scalar product of $\lambda$ and $\mathbf A$.


Thus $\mathbf C = \sqbrk c_{m n}$ is the $m \times n$ matrix composed of the product of $\lambda$ with the corresponding elements of $\mathbf A$.


Commutative Matrix Product

Definition:Commutative Matrix Product

Kronecker Product

Also known as matrix direct product:


Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{p q}$ be matrices.

The Kronecker product of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \otimes \mathbf B$ and is defined as the block matrix:

$\mathbf A \otimes \mathbf B = \begin{bmatrix} a_{11} \mathbf B & a_{12} \mathbf B & \cdots & a_{1n} \mathbf B \\ a_{21} \mathbf B & a_{22} \mathbf B & \cdots & a_{2n} \mathbf B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} \mathbf B & a_{m2} \mathbf B & \cdots & a_{mn} \mathbf B \end{bmatrix}$


Writing this out in full:

$\mathbf A \otimes \mathbf B = \begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1q} & \cdots & \cdots & a_{1n} b_{11} & a_{1n} b_{12} & \cdots & a_{1n} b_{1q} \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2q} & \cdots & \cdots & a_{1n} b_{21} & a_{1n} b_{22} & \cdots & a_{1n} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{11} b_{p1} & a_{11} b_{p2} & \cdots & a_{11} b_{pq} & \cdots & \cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \cdots & a_{1n} b_{pq} \\ \vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\ a_{m1} b_{11} & a_{m1} b_{12} & \cdots & a_{m1} b_{1q} & \cdots & \cdots & a_{mn} b_{11} & a_{mn} b_{12} & \cdots & a_{mn} b_{1q} \\ a_{m1} b_{21} & a_{m1} b_{22} & \cdots & a_{m1} b_{2q} & \cdots & \cdots & a_{mn} b_{21} & a_{mn} b_{22} & \cdots & a_{mn} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{m1} b_{p1} & a_{m1} b_{p2} & \cdots & a_{m1} b_{pq} & \cdots & \cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \cdots & a_{mn} b_{pq} \end{bmatrix}$


Thus, if:

$\mathbf A$ is a matrix with order $m \times n$
$\mathbf B$ is a matrix with order $p \times q$

then $\mathbf A \otimes \mathbf B$ is a matrix with order $m p \times n q$.


Hadamard Product

Also known as Matrix Entrywise Product or Schur Product:


Let $\struct {S, \cdot}$ be an algebraic structure.

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $S$.

Let $\mathbf B = \sqbrk b_{m n}$ be an $m \times n$ matrix over $S$.


The Hadamard product of $\mathbf A$ and $\mathbf B$ is written $\mathbf A \circ \mathbf B$ and is defined as follows:

$\mathbf A \circ \mathbf B := \mathbf C = \sqbrk c_{m n}$

where:

$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} \cdot_R b_{i j}$


Frobenius Inner Product

Definition:Frobenius Inner Product

Cracovian

Definition:Cracovian