[1] Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = BTB = A (for real-valued matrices, where BT is the transpose of B).
This distinct meaning is discussed in Positive definite matrix § Decomposition.
[2] Thus Minus I2 also has a square root, for example: which can be used to represent the imaginary unit
Notice that some ideas from number theory do not carry over to matrices: The square root of a nonnegative integer must either be another integer or an irrational number, excluding non-integer rationals.
Contrast that to a matrix of integers, which can have a square root whose entries are all non-integer rational numbers, as demonstrated in some of the above examples.
[3] The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root of A is the same as the rank of A.
[3] The operation of taking the principal square root is continuous on this set of matrices.
[4] These properties are consequences of the holomorphic functional calculus applied to matrices.
[5][6] The existence and uniqueness of the principal square root can be deduced directly from the Jordan normal form (see below).
If the diagonal elements of D are real and non-negative then it is positive semidefinite, and if the square roots are taken with the (+) sign (i.e. all non-negative), the resulting matrix is the principal root of D. A diagonal matrix may have additional non-diagonal roots if some entries on the diagonal are equal, as exemplified by the identity matrix above.
) and at most one of its diagonal entries is zero, then one upper triangular solution of the equation
increasing from 1 to n-1 as: If U is upper triangular but has multiple zeroes on the diagonal, then a square root might not exist, as exemplified by
can be diagonalized as VDV−1, where D has principal square root giving the square root When A is symmetric, the diagonalizing matrix V can be made an orthogonal matrix by suitably choosing the eigenvectors (see spectral theorem).
are positive reals, which means the resulting matrix is the principal root of
To see that any complex matrix with positive eigenvalues has a square root of the same form, it suffices to check this for a Jordan block.
Substituting N for z, only finitely many terms will be non-zero and S = √λ (I + a1 N + a2 N2 + ⋯) gives a square root of the Jordan block with eigenvalue √λ.
Any other square root T with positive eigenvalues has the form T = I + M with M nilpotent, commuting with N and hence L. But then 0 = S2 − T2 = 2(L − M)(I + (L + M)/2).
Since L and M commute, the matrix L + M is nilpotent and I + (L + M)/2 is invertible with inverse given by a Neumann series.
Hence L = M. If A is a matrix with positive eigenvalues and minimal polynomial p(t), then the Jordan decomposition into generalized eigenspaces of A can be deduced from the partial fraction expansion of p(t)−1.
By virtue of Gelfand formula, that condition is equivalent to the requirement that the spectrum of
The iteration is defined by As this uses a pair of sequences of matrix inverses whose later elements change comparatively little, only the first elements have a high computational cost since the remainder can be computed from earlier elements with only a few passes of a variant of Newton's method for computing inverses, With this, for later values of k one would set
On the other hand, as Denman–Beavers iteration uses a pair of sequences of matrix inverses whose later elements change comparatively little, only the first elements have a high computational cost since the remainder can be computed from earlier elements with only a few passes of a variant of Newton's method for computing inverses (see Denman–Beavers iteration above); of course, the same approach can be used to get the single sequence of inverses needed for the Babylonian method.
However, unlike Denman–Beavers iteration, the Babylonian method is numerically unstable and more likely to fail to converge.
[citation needed] According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T1/2 such that T1/2 is itself positive and (T1/2)2 = T. The operator T1/2 is the unique non-negative square root of T. [citation needed] A bounded non-negative operator on a complex Hilbert space is self adjoint by definition.
If T is a non-negative operator on a finite-dimensional Hilbert space, then all square roots of T are related by unitary transformations.
Indeed, take B = T1/2 to be the unique non-negative square root of T. If T is strictly positive, then B is invertible, and so U = AB−1 is unitary: 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 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.
By Choi's result, a linear map is completely positive if and only if it is of the form where k ≤ nm.
and Σ pi = 1, the set is said to be an ensemble that describes the mixed state ρ.
For instance, suppose The trace 1 condition means Let and vi be the normalized ai.