Resultados del cálculo
1. ¿Qué es una raíz cuadrada matricial?
A matrix \(X\) is called a square root of a square matrix \(A\) if
In general a matrix may have zero, one, or many square roots (over the reals or complex numbers). For real symmetric positive definite matrices there exists a unique symmetric positive definite square root, called the principal square root, usually denoted \(A^{1/2}\).
2. Cómo calcular una raíz cuadrada matricial: métodos comunes
- Diagonal (or diagonalizable) matrices: If \(A = VDV^{-1}\) with \(D=\operatorname{diag}(d_i)\), then one square root is \[ X = V\,\operatorname{diag}(\pm\sqrt{d_i})\,V^{-1}, \] provided the \(d_i\) admit square roots (e.g. \(d_i>0\) for real principal roots).
- Symmetric positive definite case: if \(A\) is symmetric positive definite and \(A=V D V^{T}\) orthogonally diagonalizes, the principal (symmetric) square root is \[ A^{1/2} = V \operatorname{diag}(\sqrt{d_i}) V^{T}. \]
- Schur method (numerical): compute real Schur \(A = Q T Q^{T}\) with \(T\) quasi-triangular, compute a square root \(S\) of \(T\) blockwise, then set \(X = Q S Q^{T}\).
- Matrix Newton iteration (numerical): \[ X_{k+1} = \tfrac{1}{2}\left(X_k + X_k^{-1} A\right), \] starting from a suitable \(X_0\), converges quadratically to a square root under conditions.
3. Ejemplos resueltos
Ejemplo 1 - matriz diagonal (fácil)
Since \(A\) is diagonal, take square roots of diagonal entries (principal positive roots):
Check: \(\begin{pmatrix}2 &0\\0&3\end{pmatrix}^2 = \begin{pmatrix}4&0\\0&9\end{pmatrix}.\)
Ejemplo 2 - matriz simétrica definida positiva 2×2 (diagonalización)
Let
Compute eigen-decomposition. Eigenvalues are \(\lambda_1=9,\ \lambda_2=1\); orthonormal eigenvectors form
Raíz cuadrada principal (usa raíces positivas):
Carrying out the multiplication (since \(V\) is orthogonal):
Check: \(\begin{pmatrix}2&1\\1&2\end{pmatrix}^2=\begin{pmatrix}5&4\\4&5\end{pmatrix}=A.\)
4. Errores comunes
- Treating square root element-wise: \(A^{1/2}\) is not generally \((\sqrt{a_{ij}})\).
- Assuming existence for arbitrary matrices: not every real matrix has a real square root (or a symmetric real square root). Check eigenvalues and Jordan blocks.
- Using diagonalization without checking: diagonalization requires a full set of eigenvectors; if \(A\) is not diagonalizable use Schur/Jordan methods.
- Mixing signs: eigenvalues have two square roots \(\pm\sqrt{\lambda}\); principal square root uses the positive branch for positive eigenvalues.
- Neglecting numerical stability: computing square roots for nearly defective matrices requires stable algorithms (Schur + careful block solves or Newton iteration).
5. Practice problems
Try these problems. Click Mostrar respuesta to reveal the (principal) square root.
Ejercicio 1
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Ejercicio 2
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(See worked Example 2.)
Ejercicio 3
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This is the principal symmetric square root (use positive root \(\sqrt{3}\)).
Ejercicio 4
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Caso diagonal: toma raíces cuadradas positivas en la diagonal.
Para estudiar mejor la raíz cuadrada de una matriz, también te pueden servir la calculadora de rango, la calculadora SVD y la calculadora RREF.
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