Square root of a matrix: Information from Answers.com

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

1 Square roots of positive operators
- 1.1 Unitary freedom of square roots
- 1.2 Some applications
2 Computing the matrix square root
- 2.1 By diagonalization
- 2.2 Denman–Beavers square root iteration
3 See also
4 References

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 an 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:

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

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

$\Phi(A) = \sum_i ^k V_i A V_i^*$

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.

Computing the matrix square root

One can also define the square root of a square matrix A that is not necessarily positive-definite. A matrix B is said to be a square root of A if the matrix product B · B is equal to A.

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 $D = V - 1 A V$ 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, $A = V D V - 1$ , and hence the square root of A is

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

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.

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

$\begin{align} Y_{k+1} &= \tfrac12 (Y_k + Z_k^{-1}), \\ Z_{k+1} &= \tfrac12 (Z_k + Y_k^{-1}). \end{align}$

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)

References

Cheng, Sheung Hun; Higham, Nicholas J.; Kenney, Charles S.; Laub, Alan J. (2001), "Approximating the Logarithm of a Matrix to Specified Accuracy", SIAM Journal on Matrix Analysis and Applications 22 (4): 1112–1125, doi:10.1137/S0895479899364015, http://www.eeweb.ee.ucla.edu/publications/journalAlanLaubajlaub_simax22(4)_2001.pdf
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

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Square root of a matrix

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