Abstract
Here we introduce the concept ofanalytic function of a square matrix and methods to compute it. We illustrate the concept with a number of examples pertaining to 2 ×2 matrices that can be handled with longhand calculations. For symmetric matrices, we introduce the Mohr circle to compute not only their eigenvalues and eigenvectors, but also their analytic functions. Moreover, we include shortcuts applicable to specific types of matrices, e.g., matrices with simple structures, with, e.g., a limited number of non-zero entries.
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Notes
- 1.
The structure of Λ is so frequent in system theory that it bears a name, Vandermonde matrix. Scientific software provides means to create a Vandermonde matrix by entering, in general, only the name (computer algebra) or the value of the argument λ and the dimension n of the square matrix.
- 2.
Invariance means that, under a change of vector basis, the trace does not change. More precisely, a quantity is invariant when it follows certain rules under a change of frame. A scalar is invariant when it does not change under a change of frame.
- 3.
A similarity transformation occurs wherever a change of variable y = Lx is introduced. Similarity transformations are studied in basic linear-algebra courses.
References
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Angeles, J. (2011). Matrix Functions. In: Dynamic Response of Linear Mechanical Systems. Mechanical Engineering Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1027-1_9
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DOI: https://doi.org/10.1007/978-1-4419-1027-1_9
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