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Square root of a matrix

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	Topics Square root of a matrix Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , the square root of a matrix extends the notion of square rootSquare root In mathematics, a square root of a number x is a number whose square is x.... from numbers to matricesMatrix (mathematics) Overview In mathematics, a matrix is a rectangular table of numbers or, more generally, a table consisting of abstract quantities tha... . Square roots of positive operators In linear algebraLinear algebra Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces , linear transformations, and... and operator theoryOperator theory In mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their pro... , given a bounded positive semidefinite operator T on a complex Hilbert space, B is a square root of T if T = BB. According to the spectral theoremSpectral theorem In mathematics, particularly linear algebra and functional analysis, the spectral theorem* is any of a number of results abou... , the continuous functional calculusContinuous functional calculus In mathematics, the continuous functional calculus of operator theory and C-algebra theory allows applications of continuou... 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 positive square root* of T. A bounded positive operator on a complex Hilbert space is self adjoint. So T = (T^½)* T^½. Conversely, it is trivially true that every operator of the form BB* is positive. Therefore, an operator T is positive if and only ifIf and only if In logic and fields that rely on it, such as mathematics and philosophy, "if and only if" is a logical connective between s... T = BB* 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 n × n positive matrix, all square roots of T are related by unitary transformations. More precisely, if T = AA* = BB, then there exists an unitaryUnitary matrix In mathematics, a unitary matrix* is an n by n complex matrix U satisfying the condition... U s.t. A = BU. We now verify this claim. Take B = T^½ to be the unique positive square root of T. It suffices to show that, for any square root A of T, A = UB. Let _{1 = i = n}, and _{1 = i = n} be the set of column vectors of A and B, respectively. By the construction of the square root, _{1 = i = n} is the set of, not necessarily normalized, eigenvectors of a Hermitian matrix, therefore an orthogonal basis for Cⁿ. Notice we include the eigenvectors corresponding to eigenvalue 0 when appropriate. The argument below hinges on the fact that _{1 = i = n} is linearly independent and spans Cⁿ. The equality AA* = BB* can be rewritten as By completeness of , for all j, there exists n scalars, by appending zeros if necessary, _{1 = i = n} s.t. i.e. We need to show the matrix U = [u_{i j}] is unitary. Compute directly So By assumption, Now because is a basis of Cⁿ, the set is a basis for n × n matrices. For linear spaces in general, the expression of an element in terms of a given basis is unique. Therefore In other words, the n column vectors of U is an orthonormal set and the claim is proved. The argument extends to the case where A and B are not necessarily square, provided one retains the assumption is linearly independent. In that case, the existence of the rectangular matrix U = [u_{i j}] follows from the relation rather than the completeness of . The conclusion UU* = I still holds. In general, B is n × m for some m = n. A is n × k where m = k = n. U is a m × k partial isometryFacts About Partial isometry In functional analysis a partial isometry is a linear map W between Hilbert spaces H and K such that the restric... . (By a partial isometry, we mean a rectangular matrix U with UU* = I.) Some applications The unitary freedom of square roots has applications in linear algebra. Kraus operators By Choi's result, a linear map is completely positive if and only if it is of the form where k = nm. Let ? C^{n × n} be the n² elementary matrix units. The positive matrix is called the Choi matrix of F. The Kraus operators correspond to the, not necessarily square, square roots of M_F: For any square root B of M_F, 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 In quantum physics, a density matrix for a n-level quantum system is a n × n complex matrix ? that is positive semidefinite with trace 1. If ? can be expressed as where ? p_i = 1, the set is said to be an ensemble that describes the mixed state ?. Notice is not required to be orthogonal. Different ensembles describing the state ? are related by unitary operators, via the square roots of ?. For instance, suppose The trace 1 condition means Let and v_i be the normalized a_i. We see that gives the mixed state ?. Operators on Hilbert space In general, if A, B are closed and densely defined operators on a Hilbert space H, and AA* = BB, then A = UB* where U is a partial isometry. 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 matrixDiagonal matrix In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero.... 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 is a diagonal matrixDiagonal matrix Overview In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero.... . 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, , and hence the square root of A is This approach works only for diagonalizable matricesDiagonalizable matrix In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e.... . For non-diagonalizable matrices one can calculate the Jordan normal formJordan normal form In linear algebra, Jordan normal form... followed by a series expansion, similar to the approach described in logarithm of a matrixLogarithm of a matrix Summary In mathematics, the logarithm of a matrix is a generalization of the scalar logarithm to matrices.... . 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 matrixIdentity matrix In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and ... . The iteration is defined by The matrix converges quadratically to the square root A^1/2, while converges to its inverse, A^-1/2 (; ) See also Holomorphic functional calculusHolomorphic functional calculus In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions....