Definition:Matrix Scalar Product

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Definition

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 $\mathbf A$.


Ring

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

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

Let $\lambda \in R$ be any element of $R$.


The scalar product of $\lambda$ and $\mathbf A$ is defined as follows.

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

Then:

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


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


Scalar

The element $\lambda$ of the underlying structure of $\map \MM {m, n}$ is known as a scalar.


Examples

Arbitrary Example

The matrix scalar product of the matrix $\begin {pmatrix} a & b \\ c & d \end {pmatrix}$ by the scalar $k$ works as follows:

$k \begin {pmatrix} a & b \\ c & d \end {pmatrix} = \begin {pmatrix} k a & k b \\ k c & k d \end {pmatrix}$


Also see

  • Results about matrix scalar product can be found here.


Sources

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