矩阵的幂
假设我们有一个Hermitian矩阵a，该> a 已知具有倒数。
我知道zgetrf和zgetri Lapack库中的子例程可以计算逆矩阵。
Lapack或Blas库中是否可以直接计算a^{ - 1/2}或任何其他方法来计算a^{ - 1/2}？
Suppose we have a Hermitian matrix A that is known to have an inverse.
I know that ZGETRF and ZGETRI subroutines in LAPACK library can compute the inverse matrix.
Is there any subroutine in LAPACK or BLAS library can calculate A^{-1/2} directly or any other way to compute A^{-1/2}?
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您可以按照与求矩阵指数类似的过程对矩阵求幂：
v_i和相应的特征值e_i。{e_i}^{-1/2}。{e_i}^{-1/2}、特征向量为v_i的矩阵。值得注意的是，正如此处所述，这个问题没有唯一的解决方案。在上面的步骤 2 中，
{e_i}^{-1/2}和-{e_i}^{-1/2}都会得出有效的解决方案，因此N*N矩阵A将至少有2^N矩阵B，使得B^{ -2}=A。如果任何特征值是退化的，那么将存在连续的有效解空间。You can raise a matrix to a power following a similar procedure to taking the exponential of a matrix:
v_iand corresponding eigenvaluese_i.{e_i}^{-1/2}.{e_i}^{-1/2}and whose eigenvectors arev_i.It's worth noting that, as described here, this problem does not have a unique solution. In step 2 above, both
{e_i}^{-1/2}and-{e_i}^{-1/2}will lead to valid solutions, so anN*NmatrixAwill have at least2^NmatricesBsuch thatB^{-2}=A. If any of the eigenvalues are degenerate then there will be a continuous space of valid solutions.