# Trigonometric Functions of Commutative Matrices

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An exponential or trigonometric function of a matrix is defined in terms of the Taylor series expansion of [2]:

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Contributed by: D. Meliga, A. Ratti and S. Z. Lavagnino (August 2019)

Additional contribution by: L. Lavagnino

Open content licensed under CC BY-NC-SA

## Details

Snapshot 1: initial matrices used for all the operations

Snapshot 2: trigonometric identity applied to the matrix

Snapshot 3: trigonometric sum to product identity applied to and matrices with an insufficient degree of the Taylor series

Snapshot 4: Baker–Campbell–Hausdorff formula applied to and matrices; as they commute, the formula is simplified

Snapshot 5: trace versus degree plot of the two trigonometric sums to product identity matrices; the two plots overlap when the degree is high enough and the identity is correct

References

[1] S. Lipschutz and M. L. Lipson, *Linear Algebra*, 4th ed., New York: McGraw-Hill, 2009.

[2] Wikipedia. "Trigonometric Functions of Matrices." (Jun 24, 2019) en.wikipedia.org/wiki/Trigonometric_functions_of_matrices.

[3] Wikipedia. "Baker–Campbell–Hausdorff formula." (Aug 9, 2019) en.wikipedia.org/wiki/Baker% E2 %80 %93 Campbell % E2 %80 %93 Hausdorff_formula.

## Snapshots

## Permanent Citation