Matrix Arccosine: Definition, Formulas, Examples, and Practice

This article explains the matrix arccosine function \(\arccos(A)\), shows definitions and practical formulas, includes two worked examples, lists common pitfalls, and provides four practice problems with collapsible answers. It is intended as companion content for a Matrix Arccosine Calculator.

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Definition and Formula

The matrix arccosine of a square matrix \(A\), denoted \(\arccos(A)\), is a matrix function acting as a (local) inverse of the matrix cosine:

$$ \cos(\arccos(A)) = A $$

A convenient relationship is:

$$ \arccos(A) \;=\; \tfrac{\pi}{2}I \;-\; \arcsin(A), $$

so one can compute \(\arccos(A)\) by computing \(\arcsin(A)\) (for a chosen branch) and subtracting from \(\tfrac{\pi}{2}I\). The matrix arcsine admits the Taylor series (for small \(\|A\|\)):

$$ \arcsin(A) = A + \tfrac{1}{6}A^{3} + \tfrac{3}{40}A^{5} + \tfrac{5}{112}A^{7} + \cdots, $$

thus

$$ \arccos(A) = \tfrac{\pi}{2}I - \Big(A + \tfrac{1}{6}A^{3} + \tfrac{3}{40}A^{5} + \cdots\Big). $$

In practice, numerical and stable methods use diagonalization, Jordan/Schur decompositions, or tailored matrix-function routines rather than naive series truncation.

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Worked Example 1 (Diagonal scalar matrix)

Let

$$ A = \begin{pmatrix} 0.2 & 0 \\[6pt] 0 & 0.2 \end{pmatrix}. $$

Because \(A = 0.2 I\), apply scalar arccos element-wise:

$$ \arccos(A) = \arccos(0.2)\, I. $$

Numerically, \(\arccos(0.2)\approx 1.369438\) (radians), so

$$ \arccos(A) \approx 1.369438 \begin{pmatrix}1 & 0\\[6pt]0 & 1\end{pmatrix}. $$

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Worked Example 2 (Nilpotent matrix — series truncation)

Let

$$ B = \begin{pmatrix}0 & 1\\[6pt]0 & 0\end{pmatrix}. $$

Note \(B^2 = 0\). Using relation \(\arccos(B)=\tfrac{\pi}{2}I-\arcsin(B)\) and the fact \(\arcsin(B)=B\) (series truncates because \(B^3=0\)), we get:

$$ \arccos(B) = \tfrac{\pi}{2}I - B = \begin{pmatrix}\tfrac{\pi}{2} & -1\\[6pt]0 & \tfrac{\pi}{2}\end{pmatrix}. $$

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Common Mistakes and Tips

Common mistake	Tip / correct approach
Applying scalar arccos element-wise to a non-diagonal matrix	Only valid for diagonal matrices or after diagonalization/similar transform that diagonalizes A.
Ignoring branch choices of inverse trig functions	Be explicit which branch of \(\arccos\) (principal value) you use; eigenvalues near branch cuts require care.
Naively truncating series for large-norm matrices	Use Schur or diagonalization methods for stability or scale-and-square approaches.
Assuming \(\arccos(A)\) always exists	Matrix function inverse exists only when matrix lies in the range of \(\cos\) on chosen domain; check spectrum and convergence.

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Practice Problems (Compute \(\arccos(A)\))

The exercises below are suitable for hand computation when possible or for use with your Matrix Arccosine Calculator. Matrices are visible; click "Show Answer" to reveal solutions.

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