Matrix Arcsine: Definition, Examples, and Practice

This article explains the matrix arcsine function (ᵔarcsinᵔ), gives formulas and worked examples, summarizes common mistakes, and provides practice problems with collapsible answers. Suitable as companion content for a Matrix Arcsine Calculator.

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Definition and Formula

The matrix arcsine of a square matrix \(A\), denoted \(\arcsin(A)\), is the matrix function that serves as an inverse to the matrix sine on an appropriate domain:

$$ \sin(\arcsin(A)) = A \quad \text{(for matrices in the domain of the inverse)} $$

One practical way to compute \(\arcsin(A)\) is via the power series (Taylor series) for the scalar arcsine applied to the matrix:

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

The scalar series converges for \(|x| \le 1\) (with endpoint behavior), and for matrices the series converges if the chosen matrix norm satisfies \(\|A\| < 1\). In practice, numerical methods, diagonalization, Jordan decomposition or Schur-based methods may be used for stability and convergence.

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Properties

• \(\arcsin(A)\) is defined only when a suitable inverse of \(\sin\) exists for that matrix (or on a chosen branch).
• If \(A\) is diagonalizable with \(A = V \, \mathrm{diag}(\lambda_i)\, V^{-1}\), then \(\arcsin(A) = V \,\mathrm{diag}(\arcsin(\lambda_i))\, V^{-1}\) when each \(\lambda_i\) lies in the domain where scalar arcsin is defined.
• If \(A\) is nilpotent (e.g. \(A^2=0\)), higher powers vanish and the series truncates.

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Worked Example 1 (Diagonal scalar matrix)

Let

$$ A = \begin{pmatrix} 0.2 & 0 \\[4pt] 0 & 0.2 \end{pmatrix}. $$

Because \(A = 0.2 I\), the arcsine is applied element-wise:

$$ \arcsin(A) = \arcsin(0.2)\, I $$

So the result is:

$$ \arcsin(A) = \arcsin(0.2)\begin{pmatrix}1 & 0\\[4pt]0 & 1\end{pmatrix}. $$

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Worked Example 2 (Nilpotent matrix)

Let

$$ B = \begin{pmatrix}0 & 1\\[4pt]0 & 0\end{pmatrix}. $$

Note \(B^2 = 0\). Using the power series for \(\arcsin\), all terms beyond the linear term vanish, so

$$ \arcsin(B) = B. $$

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Common Mistakes and Tips

Common mistake	Tip / correct approach
Applying scalar arcsin element-wise to a non-diagonal matrix	Only valid for diagonal matrices or when matrix is brought to diagonal form via similarity transform.
Ignoring convergence conditions of the power series	Ensure \(\|A\|<1\) for safe use of the series, or use diagonalization/Schur methods for larger norms.
Truncating series too early for non-nilpotent matrices	Keep enough terms for desired accuracy or use a library routine that adapts truncation.
Forgetting branch choices for inverse trig functions	Be explicit about which branch of arcsin is used when eigenvalues are near branch cuts.

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Practice Problems (Compute \(\arcsin(A)\))

The exercises below are designed for calculator use or hand computation. Matrix expressions are shown; click "Show Answer" to reveal the solution.

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