Scalar multiplication

Not to be confused with scalar product.

Scalar multiplication of a vector by a factor of 3 stretches the vector out.

The scalar multiplications −a and 2a of a vector a

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra^[1]^[2]^[3] (or more generally, a module in abstract algebra^[4]^[5]). In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and must be distinguished from 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 scalar multiplication is a function from K × V to V. The result of applying this function to c in K and v in V is denoted cv.

Properties

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

Additivity in the scalar: (c + d)v = cv + dv;
Additivity in the vector: c(v + w) = cv + cw;
Compatibility of product of scalars with scalar multiplication: (cd)v = c(dv);
Multiplying by 1 does not change a vector: 1v = v;
Multiplying by 0 gives the zero vector: 0v = 0;
Multiplying by −1 gives the additive inverse: (−1)v = −v.

Here + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field.

Interpretation

Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. A geometric interpretation of scalar multiplication is that it stretches, or contracts, vectors by a constant factor.

As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field.

When V is Kⁿ, scalar multiplication is equivalent to multiplication of each component with the scalar, and may be defined as such.

The same idea applies if K is a commutative ring and V is a module over K. K can even be a rig, but then there is no additive inverse. If K is not commutative, the distinct operations left scalar multiplication cv and right scalar multiplication vc may be defined.

Scalar multiplication of matrices

Main article: Matrix (mathematics)

The left scalar multiplication of a matrix $A$ with a scalar $λ$ gives another matrix $λ A$ of the same size as $A$ . The entries of $λ A$ are defined by

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

explicitly:

\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 $A$ with a scalar $λ$ is defined to be

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

explicitly:

\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, the real or complex number field, these two multiplications are the same, and are simply called scalar multiplication. However, for matrices over a more general ring that are not commutative, such as the quaternions, they may not be equal.

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 quaternion 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 the quaternion units. The non-commutativity of quaternion multiplication prevents the transition of changing $ij = + k$ to $ji = - k$ .

References

↑ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4.
↑ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0-03-010567-6.
↑ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN 0-387-98258-2.
↑ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
↑ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

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