# Scalar Multiplication

In mathematics , scalar multiplication is one of the basic operations of a decidable linear algebra in vector space (or more generally a module in abstract algebra   ). In general geometric contexts, scalar multiplications of a real Euclidean vector by a positive real number multiplies the vector ‘s magnitude—without changing its direction. The term ” scalar ” itself derives from this usage: a scalar is one that scales Vector scalar multiplication is the multiplications of a vector by a scalar (where the product is a vector), and is to be distinguished from the inner product of two vectors (where the product is a scalar).

## Definition

In general, if K is a field and V is a vector space over K , then the scalar multiplications is a function from K × V to V. The result of applying this function to k in k and v in v is denote k v . 

### irtue

Scalar multiplication obeys the following rules ( vector in boldface ) :

• Additivity in a scalar : ( c + d ) v = v + v ;
• Additiveness in a vector: c ( v + w ) = v + w ;
• Multiplying by 1 does not change any vector: 1 v = v ;
• Compatibility of scalar product with scalar multiplications: ( cd ) v = c ( v );
• Multiplying by 0 gives the zero vector : 0 v = 0 ;
• Multiplying by −1 gives the additive inverse : (−1) v = – v .

Here, + is the addition either in the field or in the vector space, as appropriate ; And 0 is either additive identity. Juxposition indicates either scalar multiplications or multiplication operations in the field.

## Explanation

Scalar multiplication can be viewed as an external binary operation or an action of a field on a vector space . One geometric interpretation of scalar multiplications is that it expands, or contracts, vectors by a constant factor. As a result, it produces a vector of a different length in the same or opposite direction of the original vector. 
As a special case, V can be taken as K itself and then the scalar multiplications can be taken as the only multiplication in the field.

When V is n , the scalar multiplication is equal to the multiplication with the scalar of each component, and can be defined as. The same idea applies if K is a commutative ring and V is a modulus over K. K can also be a rig , but then there is no additive inverse. If K is not commutative , then separate operations left scalar multiplications c v and right scalar multiplications v c can be defined.

## scalar multiplication of a matrix

Left scalar multiplication of a matrix with a scalar gives another matrix of the same size as a . This is shown by ,  whose entries of are defined by

(\lambda \mathbf {A} )_{ij}=\lambda \left(\mathbf {A} \right)_{ij}\,,

Clearly:

\lambda \mathbf {A} =\lambda {\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}={\begin{pmatrix}\lambda A_{11}&\lambda A_{12}&\cdots &\lambda A_{1m}\\\lambda A_{21}&\lambda A_{22}&\cdots &\lambda A_{2m}\\\vdots &\vdots &\ddots &\vdots \\\lambda A_{n1}&\lambda A_{n2}&\cdots &\lambda A_{nm}\\\end{pmatrix}}\,.

Similarly, the right scalar multiplication of a matrix with a scalar is defined to be

(\mathbf {A} \lambda )_{ij}=\left(\mathbf {A} \right)_{ij}\lambda \,,

Clearly:

\mathbf {A} \lambda ={\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}\lambda ={\begin{pmatrix}A_{11}\lambda &A_{12}\lambda &\cdots &A_{1m}\lambda \\A_{21}\lambda &A_{22}\lambda &\cdots &A_{2m}\lambda \\\vdots &\vdots &\ddots &\vdots \\A_{n1}\lambda &A_{n2}\lambda &\cdots &A_{nm}\lambda \\\end{pmatrix}}\,.

When the underlying ring is commutative , for example, in the real or complex number field , these two multiplications are the same, and are simply called scalar multiplication . However, for matrices on the more general ring that are not commutative , such as quaternions , they may not be identical.

For a real scalar and matrix:

\lambda =2,\quad \mathbf {A} ={\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}
2\mathbf {A} =2{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}={\begin{pmatrix}2\!\cdot \!a&2\!\cdot \!b\\2\!\cdot \!c&2\!\cdot \!d\\\end{pmatrix}}={\begin{pmatrix}a\!\cdot \!2&b\!\cdot \!2\\c\!\cdot \!2&d\!\cdot \!2\\\end{pmatrix}}={\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}2=\mathbf {A} 2.

For quaternary scalars and matrices:

\lambda =i,\quad \mathbf {A} ={\begin{pmatrix}i&0\\0&j\\\end{pmatrix}}
i{\begin{pmatrix}i&0\\0&j\\\end{pmatrix}}={\begin{pmatrix}i^{2}&0\\0&ij\\\end{pmatrix}}={\begin{pmatrix}-1&0\\0&k\\\end{pmatrix}}\neq {\begin{pmatrix}-1&0\\0&-k\\\end{pmatrix}}={\begin{pmatrix}i^{2}&0\\0&ji\\\end{pmatrix}}={\begin{pmatrix}i&0\\0&j\\\end{pmatrix}}i\,,

where i , j , k are quadratic units. The non-commutativity of quadratic multiplication prevents the transition of ij = + k to ji = -k .