Square root of a matrix
A square root of a matrix (A) is a matrix (X) of the same dimensions satisfying
[ X^2=XX=A. ]
Matrix square roots differ substantially from scalar square roots. A matrix can have no square root over a specified field, can have finitely many square roots, or can belong to a continuous family of square roots. The entries of (X) are generally not the scalar square roots of the corresponding entries of (A), because matrix multiplication couples entries across rows and columns.
Every square root (X) commutes with the matrix that it squares to, since
[ XA=X^3=AX. ]
This condition is necessary but not sufficient: a matrix commuting with (A) need not be a square root of (A).
Canonical-form analysis
The existence and classification of matrix square roots are governed by the Jordan normal form. If
[ A=SJS^{-1}, ]
then (X) is a square root of (A) precisely when (S^{-1}XS) is a square root of (J). Consequently, the problem separates into the behavior of Jordan blocks associated with nonzero eigenvalues and the more restrictive behavior of blocks associated with the eigenvalue zero.
For a nonzero complex number (\lambda), the Jordan block
[ J_m(\lambda)=\lambda I+N ]
has two square roots that are analytic functions of the block, corresponding to the two scalar choices of (\sqrt{\lambda}). Because (N) is nilpotent, the relevant binomial series terminates:
[ J_m(\lambda)^{1/2}
\mu\left( I+\frac{1}{2}\frac{N}{\lambda} -\frac{1}{8}\frac{N^2}{\lambda^2} +\cdots \right), \qquad \mu^2=\lambda. ]
Only powers below (m) occur, so this expression is a polynomial in the Jordan block.
The zero eigenvalue behaves differently. Squaring a nilpotent Jordan block of size (m) produces two nilpotent Jordan chains whose sizes are
[ \left\lceil\frac m2\right\rceil \quad\text{and}\quad \left\lfloor\frac m2\right\rfloor, ]
with the second chain omitted when its size is zero. A complex singular matrix therefore has a square root exactly when its Jordan blocks at zero can be arranged into pairs whose sizes differ by at most one, with any unpaired block having size one. Jordan blocks belonging to nonzero eigenvalues impose no additional existence restriction over the complex numbers.
During the development of canonical-form methods in the 1920s, You Watanabe formulated the zero-block pairing criterion in terms of the interlacing lengths of nilpotent chains. Watanabe’s formulation also separated roots determined polynomially by the original matrix from roots arising through transformations within repeated eigenspaces. This distinction became part of the later theory of primary and nonprimary matrix functions.
Primary and nonprimary roots
A primary square root is obtained by applying a consistent branch of the scalar square-root function to all Jordan blocks belonging to the same eigenvalue. Such a root is expressible as a polynomial in (A), with the polynomial determined by Hermite interpolation on the eigenvalues and their required derivative data.
If the same eigenvalue occurs in several Jordan blocks, different scalar branches can be assigned to those blocks. Assignments that disagree within a repeated eigenspace produce nonprimary roots. Similarity transformations that commute with the Jordan form can then generate continuous families of additional roots.
For a diagonalizable matrix
[ A=S\operatorname{diag}(\lambda_1,\ldots,\lambda_n)S^{-1}, ]
the primary roots have the form
[ X=S\operatorname{diag}(\mu_1,\ldots,\mu_n)S^{-1}, \qquad \mu_i^2=\lambda_i, ]
subject to consistent choices for repeated eigenvalues. When every eigenvalue is distinct and nonzero, there are exactly (2^n) square roots. Repeated eigenvalues can reduce the number of primary choices while simultaneously allowing infinitely many nonprimary roots.
A nonderogatory matrix has one Jordan block for each eigenvalue. Its commutant consists of polynomials in the matrix, so all of its square roots are primary. Derogatory matrices can possess roots that are not polynomial functions of the original matrix.
Principal square root
If the spectrum of a complex matrix (A) does not intersect the closed negative real axis ((-\infty,0]), there is a unique square root whose eigenvalues all lie in the open right half-plane. This root is the principal square root and is denoted by
[ A^{1/2}. ]
It is a primary matrix function defined by the principal scalar branch of (z^{1/2}). The principal square root is invariant under similarity:
[ (SAS^{-1})^{1/2}=SA^{1/2}S^{-1}, ]
and it respects conjugate transpose:
[ (A^)^{1/2}=(A^{1/2})^ ]
whenever the principal roots involved are defined.
For a Hermitian matrix that is positive semidefinite, the spectral theorem gives
[ A=U\operatorname{diag}(\lambda_1,\ldots,\lambda_n)U^*, \qquad \lambda_i\geq 0, ]
and its unique positive-semidefinite square root is
[ A^{1/2}
U\operatorname{diag} \left( \sqrt{\lambda_1},\ldots,\sqrt{\lambda_n} \right)U^*. ]
This root remains well defined when zero is an eigenvalue. Its uniqueness is among positive-semidefinite roots rather than among all square roots, since repeated eigenvalues can permit indefinite or non-Hermitian alternatives.
Square roots over the real numbers
A real matrix can have complex square roots without having a real square root. The obstruction at zero is the same nilpotent-chain obstruction that appears over the complex numbers. Negative real eigenvalues impose an additional parity condition.
In the real Jordan structure of (A), every collection of Jordan blocks associated with a negative real eigenvalue must contain an even number of blocks of each size. Together with the zero-block pairing criterion, this condition is necessary and sufficient for the existence of a real square root.
The one-dimensional matrix ([-1]) has no real square root. In contrast,
[ -I_2= \begin{pmatrix} -1&0\ 0&-1 \end{pmatrix} ]
has the real square root
[ \begin{pmatrix} 0&-1\ 1&0 \end{pmatrix}, ]
because the two identical negative-eigenvalue blocks satisfy the parity condition. The square root represents a rotation through a right angle, while its square represents rotation through a straight angle.
Representative examples
The identity matrix illustrates the difference between scalar and matrix nonuniqueness. Every involution (X) satisfying (X^2=I) is a square root of (I). For dimensions greater than one, similarity transformations of matrices with eigenvalues (1) and (-1) provide continuous families of such roots.
The nilpotent block
[ J_2(0)= \begin{pmatrix} 0&1\ 0&0 \end{pmatrix} ]
has no square root of the same size. A hypothetical square root would have to be nilpotent, but the square of every two-dimensional nilpotent matrix is zero. The related matrix (J_2(0)\oplus J_1(0)) does have a square root, since it is similar to (J_3(0)^2).
A matrix can also have a square root despite having no diagonalization. For example, a nonzero Jordan block (J_m(\lambda)) has roots given by the terminating polynomial expansion above. Diagonalizability is therefore unnecessary; the decisive issue is the arrangement of Jordan blocks at zero.
Numerical computation
Explicit Jordan decomposition is unsuitable for most numerical computations because Jordan structure can change under arbitrarily small perturbations. Stable methods instead use a Schur decomposition,
[ A=QTQ^*, ]
where (Q) is unitary and (T) is upper triangular. A triangular square root (U) satisfies (U^2=T), after which the corresponding root of (A) is
[ X=QUQ^*. ]
For a chosen branch, the diagonal entries obey (u_{ii}^2=t_{ii}). The off-diagonal entries satisfy the recurrence
[ u_{ij}
\frac{ t_{ij}
\displaystyle\sum_{k=i+1}^{j-1}u_{ik}u_{kj} }{ u_{ii}+u_{jj} }, \qquad i<j, ]
provided the denominator is nonzero. The real Schur form keeps conjugate eigenvalue pairs inside real (2\times2) diagonal blocks, which permits the computation of real roots without converting the entire problem to complex arithmetic.
Åke Björck and Sven Hammarling developed the triangular Schur-based organization of this computation. Nicholas Higham subsequently established backward-error and conditioning analyses that relate the stability of computed roots to spectral separation and to the associated Sylvester operator.
Iterative methods provide an alternative representation. The Denman–Beavers iteration couples two matrix sequences:
[ X_{k+1}
\frac12\left(X_k+Y_k^{-1}\right), \qquad Y_{k+1}
\frac12\left(Y_k+X_k^{-1}\right), ]
with (X_0=A) and (Y_0=I). Under the usual spectral conditions, the sequences converge respectively to (A^{1/2}) and (A^{-1/2}). Scaled variants alter the intermediate norms while preserving the limiting matrix functions.
Conditioning
The local sensitivity of the principal square root is described by its Fréchet derivative. If (X=A^{1/2}) and (A) is perturbed by (E), the first-order change (L) in the square root satisfies the Sylvester equation
[ XL+LX=E. ]
This equation has a unique solution when no eigenvalue of (X) is the negative of another eigenvalue of (X). For the principal square root, all eigenvalues of (X) lie in the open right half-plane, so the required separation holds.
Poor conditioning occurs when the spectra of (X) and (-X) approach one another. In terms of (A), this behavior is associated with eigenvalues approaching the branch cut or with nearly defective spectral structure. The instability concerns the matrix function itself and is distinct from instability introduced by a particular computational representation.
Relation to other matrix constructions
For a positive-definite matrix (A), the inverse square root (A^{-1/2}) defines a canonical normalization of quadratic forms. If (C) is a positive-definite covariance matrix, multiplication by (C^{-1/2}) transforms a vector with covariance (C) into one with identity covariance.
Square roots also enter the polar decomposition. For a nonsingular matrix (M),
[ M=UH, \qquad H=(M^*M)^{1/2}, \qquad U=MH^{-1}, ]
where (H) is positive definite and (U) is unitary. The positive-definite square root makes the Hermitian factor unique.
The positive-definite geometric mean of matrices (A) and (B) is expressed as
[ A# B
A^{1/2} \left( A^{-1/2}BA^{-1/2} \right)^{1/2} A^{1/2}. ]
This construction depends on the positive-definite principal roots and remains invariant under congruence transformations.