Arcoseno matricial: definición, ejemplos y práctica

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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Definición y fórmula

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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Ejemplo resuelto 1 (matriz escalar diagonal)

Let

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

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

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

Así, el resultado es:

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

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Ejemplo resuelto 2 (matriz nilpotente)

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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Errores comunes y consejos

Common mistake	Tip / correct approach
Aplicar el arcoseno escalar elemento a elemento a una matriz no diagonal	Only valid for diagonal matrices or when matrix is brought to diagonal form via similarity transform.
Ignorar las condiciones de convergencia de la serie de potencias	Ensure \(\|A\|<1\) for safe use of the series, or use diagonalization/Schur methods for larger norms.
Truncar la serie demasiado pronto para matrices no nilpotentes	Keep enough terms for desired accuracy or use a library routine that adapts truncation.
Olvidar la elección de ramas en las funciones trigonométricas inversas	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.

El arcoseno matricial también puede estudiarse con la calculadora de rango, la calculadora SVD y la calculadora RREF.

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