Square root of a matrix

In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices.

Square root of a matrix

A matrix B is said to be a square root of A if the matrix product B · B is equal to A .

Computation by diagonalization

The square root of a diagonal matrix D is formed by taking the square root of all the entries on the diagonal. This suggests the following methods for general matrices:

An n × n matrix A is diagonalizable if there is a matrix V such that $D=V^{-1}AV$ is a diagonal matrix. This happens if and only if A has n eigenvectors which constitute a basis for Cⁿ; in this case, V can be chosen to be the matrix with the n eigenvectors as columns.

Now, $A=VDV^{-1}$, and hence the square root of A is

$A^{1/2}=VD^{1/2}V^{-1}.\,$

This approach works only for diagonalizable matrices. For non-diagonalizable matrices one can calculate the Jordan normal form followed by a series expansion, similar to the approach described in logarithm of a matrix.

Computation by Denman–Beavers square root iteration

Another way to find the square root of a matrix A is the Denman–Beavers square root iteration. Let Y₀ = A and Z₀ = I, where I is the identity matrix. The iteration is defined by

${\begin{aligned}Y_{k+1}&={\tfrac {1}{2}}(Y_{k}+Z_{k}^{-1}),\\Z_{k+1}&={\tfrac {1}{2}}(Z_{k}+Y_{k}^{-1}).\end{aligned}}$

The matrix $Y_{k}$ converges quadratically to the square root A^1/2, while $Z_{k}$ converges to its inverse, A^−1/2 (Denman & Beavers 1976; Cheng et al. 2001)

Square roots of positive operators

In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) T on a complex Hilbert space, B is a square root of T if T = B* B, where B* denotes the Hermitian adjoint of B. According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T^½ such that T^½ is itself positive and (T^½)² = T. The operator T^½ is the unique non-negative square root of T.

A bounded non-negative operator on a complex Hilbert space is self adjoint by definition. So T = (T^½)* T^½. Conversely, it is trivially true that every operator of the form B* B is non-negative. Therefore, an operator T is non-negative if and only if T = B* B for some B (equivalently, T = CC* for some C).

The Cholesky factorization is a particular example of square root.

Unitary freedom of square roots

If T is a non-negative operator on a finite dimensional Hilbert space, then all square roots of T are related by unitary transformations. More precisely, if T = AA* = BB*, then there exists a unitary U s.t. A = BU.

Indeed, take B = T^½ to be the unique non-negative square root of T. If T is strictly positive, then B is invertible, and so U = B⁻¹A is unitary:

${\begin{aligned}U^{*}U&=\left(A^{*}(B^{*})^{-1}\right)\left(B^{-1}A\right)=A^{*}(T^{-1})A\\&=A^{*}((A^{*})^{-1}A^{-1})A=I.\end{aligned}}$

If T is non-negative without being strictly positive, then the inverse of B cannot be defined, but the Moore–Penrose pseudoinverse B⁺ can be. In that case, the operator B⁺A is a partial isometry, that is, a unitary operator from the range of T to itself. This can then be extended to a unitary operator U on the whole space by setting it equal to the identity on the kernel of T. More generally, this is true on an infinite-dimensional Hilbert space if, in addition, T has closed range. In general, if A, B are closed and densely defined operators on a Hilbert space H, and A* A = B* B, then A = UB where U is a partial isometry.

Some applications

Square roots, and the unitary freedom of square roots have applications throughout functional analysis and linear algebra.

Polar decomposition

Main article: Polar decomposition

If A is an invertible operator on a finite-dimensional Hilbert space, then there is a unique unitary operator U and positive operator P such that

$A=UP;\,$

this is the polar decomposition of A. The positive operator P is the unique positive square root of the positive operator A^∗A, and U is defined by U = AP⁻¹.

If A is not invertible, then it still has a polar composition in which P is defined in the same way (and is unique). The unitary operator U is not unique. Rather it is possible to determine a "natural" unitary operator as follows: AP⁺ is a unitary operator from the range of A to itself, which can be extended by the identity on the kernel of A^∗. The resulting unitary operator U then yields the polar decomposition of A.

Kraus operators

Main article: Choi's theorem on completely positive maps

By Choi's result, a linear map

$\Phi :C^{n\times n}\rightarrow C^{m\times m}$

is completely positive if and only if it is of the form

$\Phi (A)=\sum _{i}^{k}V_{i}AV_{i}^{*}$

where k ≤ nm. Let {E_{p q}} ⊂ C^{n × n} be the n² elementary matrix units. The positive matrix

$M_{\Phi }=(\Phi (E_{pq}))_{pq}\in C^{nm\times nm}$

is called the Choi matrix of Φ. The Kraus operators correspond to the, not necessarily square, square roots of M_Φ: For any square root B of M_Φ, one can obtain a family of Kraus operators V_i by undoing the Vec operation to each column b_i of B. Thus all sets of Kraus operators are related by partial isometries.

Mixed ensembles

Main article: Density matrix

In quantum physics, a density matrix for an n-level quantum system is an n × n complex matrix ρ that is positive semidefinite with trace 1. If ρ can be expressed as

$\rho =\sum _{i}p_{i}v_{i}v_{i}^{*}$

where ∑ p_i = 1, the set

$\{p_{i},v_{i}\}\,$

is said to be an ensemble that describes the mixed state ρ. Notice {v_i} is not required to be orthogonal. Different ensembles describing the state ρ are related by unitary operators, via the square roots of ρ. For instance, suppose

$\rho =\sum _{j}a_{j}a_{j}^{*}.$

The trace 1 condition means

$\sum _{j}a_{j}^{*}a_{j}=1.$

Let

$p_{i}=a_{i}^{*}a_{i},$

and v_i be the normalized a_i. We see that

$\{p_{i},v_{i}\}\,$

gives the mixed state ρ.

See also

• Holomorphic functional calculus

Bibliography

• Cheng, Sheung Hun; Higham, Nicholas J.; Kenney, Charles S.; Laub, Alan J. (2001), "Approximating the Logarithm of a Matrix to Specified Accuracy" (PDF), SIAM Journal on Matrix Analysis and Applications, 22 (4): 1112–1125, doi:10.1137/S0895479899364015
• Denman, Eugene D.; Beavers, Alex N. (1976), "The matrix sign function and computations in systems", Applied Mathematics and Computation, 2 (1): 63–94, doi:10.1016/0096-3003(76)90020-5