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Mathematics for Physics

246 Diag

A mapping that builds a matrix with given scalar values on the main diagonal and zeros elsewhere that is used to write diagonal matrices from a sequence of scalars.

definition [d] (\(\operatorname{diag}\) = Diagonal Matrix Constructor) The \(n\times n\) matrix associated with scalars \(a_{1},\ldots,a_{n}\), having those entries on the main diagonal and zeros elsewhere:

  • \(\operatorname{diag}(a_{1},\ldots,a_{n}) = \begin{pmatrix} a_{1} & 0 & \cdots & 0 \\ 0 & a_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{n} \end{pmatrix}\) .

where

  • \(a_{1},\ldots,a_{n}\) are scalars.
  • \(n\) is the matrix size.
  • \(\operatorname{diag}(a_{1},\ldots,a_{n})\) is the corresponding diagonal matrix.

Note:

  • example: \(\eta_{\mu\nu} = \operatorname{diag}(-1,\, 1,\, 1,\, 1)\).
  • example: \((g_{ij}) = \operatorname{diag}(1,\, r^{2},\, r^{2}\sin^{2}\theta)\) in spherical coordinates.

definition [d] (Diagonal Array) An array \(A_{ij}\) whose off-diagonal entries all vanish:

  • \(A_{ij} = 0\) whenever \(i \neq j\) .

where

  • \(A_{ij}\) are the components of the array.
  • \(i, j\) are discrete indices.

Note:

  • the nonzero entries then lie only on the main diagonal \(A_{ii}\).
  • a metric with \(g_{ij} = 0\) for \(i \neq j\) is called a diagonal metric in those coordinates.

246.1 Elementary Example

246.1.1 Simple

The map \(\operatorname{diag}\) places given scalars on the main diagonal.

\[ \operatorname{diag}(2,5) = \begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix} \]

where

  • \(2\) and \(5\) are the diagonal entries.
  • off-diagonal entries are zero.

246.1.2 General

In three dimensions, three scalars determine a \(3 \times 3\) diagonal matrix.

\[ \operatorname{diag}(1,2,3) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]

\[ \operatorname{diag}(-1,1,1,1) = \eta \]

where

  • \(\eta\) illustrates a \(4 \times 4\) diagonal array used as a metric matrix.

246.2 References

  1. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — \(\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1)\).
  2. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — diagonal metric components in orthogonal coordinates.
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — diagonal matrices.