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Mathematics for Physics

261 Operator

A linear transformation from a vector space to itself that is used to transform vectors and to solve equations on that space.

definition [d] (Operator = Linear Operator = Linear Transformation = Linear Map) A mapping \(T: V \rightarrow W\) between vector spaces over the same field that preserves linear combinations:

  • \(T(\alpha\mathbf{u} + \beta\mathbf{v}) = \alpha\, T(\mathbf{u}) + \beta\, T(\mathbf{v})\) .

where

  • \(V\) and \(W\) are vector spaces over the same scalar field \(K\).
  • \(\mathbf{u}, \mathbf{v} \in V\) and \(\alpha, \beta \in K\).

definition [d] (Operator = Linear Operator = Linear Transformation = Linear Map) A mapping \(T: V \rightarrow W\) between vector spaces over the same field that satisfies, for all vectors and scalars:

  • (Additivity) \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\) .
  • (Homogeneity) \(T(\alpha\mathbf{u}) = \alpha\, T(\mathbf{u})\) .

where

  • \(V\) and \(W\) are vector spaces over the same scalar field \(K\).
  • \(\mathbf{u}, \mathbf{v} \in V\) and \(\alpha \in K\).

261.1 Examples

example 1 [d] (Matrix multiplication — Shifrin & Adams) The operator \(T(\mathbf{x}) = A\mathbf{x}\) with

  • \(A = \begin{pmatrix} 1 & 3 \\ 2 & -1 \\ 1 & 1 \end{pmatrix}\), \(\mathbf{x} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}\) .

acts as

  • \(A\mathbf{x} = \begin{pmatrix} 1 \\ 9 \\ 3 \end{pmatrix}\) .

example 2 [d] (Matrix multiplication — Kreyszig) The operator \(T(\mathbf{x}) = A\mathbf{x}\) with

  • \(A = \begin{pmatrix} 4 & 2 \\ 1 & 8 \end{pmatrix}\), \(\mathbf{x} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}\) .

acts as

  • \(A\mathbf{x} = \begin{pmatrix} 22 \\ 43 \end{pmatrix}\) .

example 3 [d] (Differentiation — Aleksandrov et al.) The operator \(D = \dfrac{d}{dx}\) with

  • input \(y = x^{2}\) .

acts as

  • \(Dy = 2x\) .

261.2 Elementary Example

261.2.1 Simple

An operator is a linear map from a space to itself. On \(\mathbb{R}^{2}\), matrix multiplication defines an operator.

\[ T(\mathbf{x}) = A\mathbf{x},\quad A = \begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix} \]

\[ T(1,0) = (1,0),\quad T(0,1) = (1,2) \]

where

  • \(T : V \rightarrow V\) is the operator.
  • \(A\) is the matrix of \(T\).

261.2.2 General

Differentiation on polynomials is an operator on an infinite-dimensional space; on the span of \(\{1,x,x^{2}\}\) it acts by

\[ D(1) = 0,\quad D(x) = 1,\quad D(x^{2}) = 2x \]

\[ D(a + bx + cx^{2}) = b + 2c x \]

where

  • \(D = \dfrac{d}{dx}\) is the differentiation operator.

261.3 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — linear operator axioms; matrix example \(A=\begin{pmatrix}4&2\\1&8\end{pmatrix}\), \(\mathbf{x}=\begin{pmatrix}3\\5\end{pmatrix}\).
  2. Hassani, S. Mathematical Physics: A Modern Introduction to Its Foundations. Springer, 2013. — linear transformation / linear map.
  3. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — linear operator / linear transformation.
  4. Shifrin, T., & Adams, M. Linear Algebra: A Geometric Approach. W. H. Freeman, 2010. — matrix example \(A=\begin{pmatrix}1&3\\2&-1\\1&1\end{pmatrix}\), \(\mathbf{x}=\begin{pmatrix}4\\-1\end{pmatrix}\).
  5. Aleksandrov, A. D., Kolmogorov, A. N., & Lavrent’ev, M. A. Mathematics: Its Content, Methods and Meaning. Dover, 1999. — differentiation example \(y=x^{2}\mapsto 2x\).
  6. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — matrix multiplication as a linear transformation.
  7. Cahill, K. Physical Mathematics. Cambridge University Press, 2019. — matrix multiplication as a linear transformation.