Square root of a matrix

Template:Short description Template:Use American English In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix Template:Mvar is said to be a square root of Template:Mvar if the matrix product Template:Math is equal to Template:Mvar.

Some authors use the name square root or the notation Template:Math only for the specific case when Template:Mvar is positive semidefinite, to denote the unique matrix Template:Mvar that is positive semidefinite and such that Template:Math (for real-valued matrices, where Template:Math is the transpose of Template:Mvar).

Less frequently, the name square root may be used for any factorization of a positive semidefinite matrix Template:Mvar as Template:Math, as in the Cholesky factorization, even if Template:Math. This distinct meaning is discussed in the article Template:Section link.

Examples

In general, a matrix can have several square roots. In particular, if $A=B^2$ then $A=(-B{)}^2$ as well.

For example, the Template:Tmath identity matrix $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ has infinitely many square roots. They are given by

$\left[\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right]$ and $\left[\begin{array}{cc}a& b\\ c& -a\end{array}\right]$

where $(a,b,c)$ are any numbers (real or complex) such that Template:Tmath.

In particular, if $(r,s,t)$ is any Pythagorean triple,Template:Efn then $\frac{1}{t}\left[\begin{array}{cc}r& s\\ s& -r\end{array}\right]$ is one of the matrix square roots of $I_2$ which happens to be symmetric and has rational entries. Thus

Template:Tmath.

Minus Template:Tmath also has square roots, for example:

Template:Tmath,

which can be used to represent the imaginary unit $i$ and hence all complex numbers using Template:Tmath real matrices; see matrix representation of complex numbers.

As with the real numbers, a real matrix may fail to have a real square root, but may have a square root with complex-valued entries. But some matrices have no square root. An example is the matrix Template:Tmath.

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.

Positive semidefinite matrices

Template:See also A symmetric real Template:Tmath matrix $A$ is called positive semidefinite if $x^{𝖳}Ax\ge 0$ for all $x\in {ℝ}^n$ (here $x^{𝖳}$ denotes the transpose of Template:Tmath, changing a column vector into a row vector).

A square real matrix $A$ is positive semidefinite if and only if $A=B^{𝖳}B$ for some matrix Template:Tmath. There can be many different such matrices Template:Tmath. A positive semidefinite matrix Template:Tmath can also have many matrices Template:Tmath such that Template:Tmath. However, Template:Tmath always has precisely one square root Template:Tmath that is both positive semidefinite and symmetric. In particular, since Template:Tmath is required to be symmetric, Template:Tmath, so the two conditions $A=BB$ or $A=B^{𝖳}B$ are equivalent.

For complex-valued matrices, the conjugate transpose $B^{\ast }$ is used instead of the transpose Template:Tmath, and the matrices Template:Tmath which are positive semidefinite over Template:Tmath are necessarily Hermitian, the latter meaning Template:Tmath.

This unique matrix is called the principal, non-negative, or positive square root (the latter in the case of a positive definite matrix).

The principal square root of a real positive semidefinite matrix is real. The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root of Template:Tmath is the same as the rank of Template:Tmath.

The operation of taking the principal square root is continuous on this set of matrices. These properties are consequences of the holomorphic functional calculus applied to matrices. The existence and uniqueness of the principal square root can be deduced directly from the Jordan normal form (see below).

Matrices with distinct eigenvalues

An Template:Math matrix with Template:Mvar distinct non-zero eigenvalues has Template:Math square roots. Such a matrix, Template:Mvar, has an eigendecomposition Template:Math, where Template:Mvar is the matrix whose columns are eigenvectors of Template:Mvar and Template:Mvar is the diagonal matrix whose diagonal elements are the corresponding Template:Mvar eigenvalues Template:Math. Thus the square roots of Template:Mvar are given by Template:Math, where Template:Math is any square root matrix of Template:Mvar, which, for distinct eigenvalues, must be diagonal with diagonal elements equal to square roots of the diagonal elements of Template:Mvar; since there are two possible choices for a square root of each diagonal element of Template:Mvar, there are Template:Math choices for the matrix Template:Math.

This also leads to a proof of the above observation, that a positive-definite matrix has precisely one positive-definite square root: a positive definite matrix has only positive eigenvalues, and each of these eigenvalues has only one positive square root; and since the eigenvalues of the square root matrix are the diagonal elements of Template:Math, for the square root matrix to be itself positive definite necessitates the use of only the unique positive square roots of the original eigenvalues.

Solutions in closed form

Template:See also If a matrix is idempotent, meaning Template:Tmath, then by definition one of its square roots is the matrix Template:Tmath itself.

Diagonal matrices

If Template:Tmath is a diagonal Template:Tmath matrix Template:Tmath, then some of its square roots are the diagonal matrices Template:Tmath where Template:Tmath.

If the diagonal elements of Template:Tmath are real and non-negative, then Template:Tmath is positive semidefinite; and if the square roots are taken with the (+) sign (i.e., all non-negative), then the resulting matrix is the principal square root of Template:Tmath.

A diagonal matrix may have additional non-diagonal square roots if some entries on its diagonal are equal, as exemplified by the identity matrix above.

Upper triangular matrices

If Template:Tmath is an upper triangular matrix (meaning its entries are $u_{i,j}=0$ for Template:Tmath) and at most one of its diagonal entries is zero, then one upper triangular solution of the equation $B^2=U$ can be found as follows.

Since the equation $u_{i,i}=b_{i,i}^2$ should be satisfied, let $b_{i,i}$ be the principal square root of the complex number Template:Tmath. This implies $b_{i,i}+b_{j,j}\ne 0$ for all Template:Tmath, because the principal square roots of complex numbers all lie on one half of the complex plane.Template:Efn This is possible with the assumption Template:Tmath for at most one Template:Tmath.

From the equation $$u_{i,j}=b_{i,i}{b}_{i,j}+b_{i,i+1}{b}_{i+1,j}+b_{i,i+2}{b}_{i+2,j}+\dots +b_{i,j-1}{b}_{j-1,j}+b_{i,j}{b}_{j,j},$$ we deduce that $b_{i,j}$ can be computed recursively for $j-i$ increasing from Template:Tmath to Template:Tmath as: $$b_{i,j}=\frac{u_{i,j}-b_{i,i+1}{b}_{i+1,j}-b_{i,i+2}{b}_{i+2,j}-\dots -b_{i,j-1}{b}_{j-1,j}}{b_{i,i}+b_{j,j}}.$$

If Template:Tmath is upper triangular but has multiple zeroes on the diagonal, then a square root might not exist, as exemplified by Template:Tmath.

Note that the diagonal entries of a triangular matrix are precisely its eigenvalues (see Triangular matrix#Properties).

By diagonalization

An Template:Math matrix Template:Mvar is diagonalizable if there is a matrix Template:Mvar and a diagonal matrix Template:Mvar such that Template:Math. This happens if and only if Template:Mvar has Template:Math eigenvectors which constitute a basis for Template:Tmath. In this case, Template:Mvar can be chosen to be the matrix with the Template:Math eigenvectors as columns, and thus a square root of Template:Mvar is

Template:Tmath,

where Template:Mvar is any square root of Template:Mvar. Indeed, $${\left(V{D}^{\frac{1}{2}}{V}^{-1}\right)}^2=V{D}^{\frac{1}{2}}\left(V^{-1}V\right)D^{\frac{1}{2}}{V}^{-1}=VD{V}^{-1}=A.$$

For example, the matrix $A=\left[\begin{array}{cc}33& 24\\ 48& 57\end{array}\right]$ can be diagonalized as Template:Math, where

$V=\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right]$ and Template:Tmath.

Template:Mvar has principal square root

Template:Tmath,

giving the square root

Template:Tmath.

When Template:Mvar is symmetric, the diagonalizing matrix Template:Mvar can be made an orthogonal matrix by suitably choosing the eigenvectors (see spectral theorem). Then the inverse of Template:Mvar is simply the transpose, so that

Template:Tmath.

By Schur decomposition

Every complex-valued square matrix Template:Tmath, regardless of diagonalizability, has a Schur decomposition given by $A=QU{Q}^{\ast }$ where $U$ is upper triangular and $Q$ is unitary (meaning Template:Tmath). The eigenvalues of $A$ are exactly the diagonal entries of Template:Tmath; if at most one of them is zero, then the following is a square root of Template:Tmath:

Template:Tmath,

where a square root $U^{\frac{1}{2}}$ of the upper triangular matrix $U$ can be found as described above.

If $A$ is positive definite, then its eigenvalues are all positive reals, so one can choose positive reals for all the diagonal entries of Template:Tmath. Hence the eigenvalues of $Q{U}^{\frac{1}{2}}{Q}^{\ast }$ are all positive reals, which means the resulting matrix is the principal root of Template:Tmath.

By Jordan decomposition

Similarly as for the Schur decomposition, every square matrix $A$ can be decomposed as $A=PJ{P}^{-1}$ where:

• Template:Tmath is invertible, and
• Template:Tmath is in Jordan normal form, that is, Template:Tmath is a block-diagonal matrix Template:Tmath where Template:Tmath are Jordan blocks (thus, Template:Tmath is also upper triangular).

To see that any real matrix with positive eigenvalues has a real matrix square root with positive eigenvalues, it suffices to check this for a Jordan block. Any such block Template:Tmath has the form Template:Tmath where:

• Template:Tmath is real positive, and
• Template:Tmath is nilpotent, that is, Template:Tmath for some positive integer Template:Tmath.

Consider the MacLaurin series $$(1+z{)}^{\frac{1}{2}}=1+(\genfrac{}{}{0ex}{}{\frac{1}{2}}{1})z+(\genfrac{}{}{0ex}{}{\frac{1}{2}}{2})z^2+\cdots .$$ Substituting Template:Tmath for Template:Tmath, one gets a finite sum, since only the Template:Tmath first terms may be non-zero. It follows that $$B^{\frac{1}{2}}=\sqrt{\lambda (I+N)}=\sqrt{\lambda }(I+(\genfrac{}{}{0ex}{}{\frac{1}{2}}{1})N+(\genfrac{}{}{0ex}{}{\frac{1}{2}}{2})N^2+\cdots +(\genfrac{}{}{0ex}{}{\frac{1}{2}}{\ell -1})N^{\ell -1}).$$ Template:Tmath is upper triangular with all eigenvalues zero, so Template:Tmath is upper triangular with all eigenvalues Template:Tmath, real positive.

Thus, Template:Tmath where Template:Tmath are upper triangular (but not necessarily Jordan blocks).

Finally, Template:Tmath.

The validity of the result may be proved by expanding the square of the result. A simpler method consists of replacing Template:Tmath with Template:Tmath, where Template:Tmath is an auxiliary real variable, mimicking the proof of Taylor's theorem for this case, and making Template:Tmath at the end.

By power series

Recall the formal power series $$(1-z{)}^{\frac{1}{2}}=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n(\genfrac{}{}{0ex}{}{\frac{1}{2}}{n})z^n,$$ which converges provided that $|z|\le 1$ (since the coefficients of the power series are summable). Plugging in $z=I-A$ into this expression yields $$A^{\frac{1}{2}}=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n(\genfrac{}{}{0ex}{}{\frac{1}{2}}{n})(I-A{)}^n,$$ provided that Template:Nobr By virtue of Gelfand formula, this condition is equivalent to the requirement that the spectrum of $A$ is contained within the closed disk Template:Tmath. This method of defining or computing $A^{\frac{1}{2}}$ is especially useful in the case where $A$ is positive semi-definite, and symmetric if real.Template:Efn Then, we have $\Vert I-\frac{A}{\Vert A\Vert }\Vert \le 1$ and therefore Template:Nobr so that $$A^{\frac{1}{2}}=\Vert A{\Vert }^{\frac{1}{2}}\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n(\genfrac{}{}{0ex}{}{\frac{1}{2}}{n}){(I-\frac{A}{\Vert A\Vert })}^n$$ defines a square root of $A$ which moreover turns out to be the unique symmetric positive semi-definite one. This method remains valid to define square roots of operators on infinite-dimensional Banach or Hilbert spaces or certain elements of (Template:Tmath) Banach algebras.

Iterative solutions

By Denman–Beavers iteration

Another way to find the square root of an Template:Tmath matrix Template:Tmath is the Denman–Beavers square root iteration.

Let Template:Tmath and Template:Tmath, where Template:Tmath is the Template:Tmath identity matrix. The iteration is defined by

$\begin{array}{cc}{Y}_{k+1}& =\frac{1}{2}(Y_k+Z_k^{-1}),\\ Z_{k+1}& =\frac{1}{2}(Z_k+Y_k^{-1}).\end{array}$

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,

Template:Tmath.

With this, for later values of Template:Tmath one would set $X_0=Z_{k-1}^{-1}$ and Template:Tmath, and then use $Z_k^{-1}=X_n$ for some small $n$ (perhaps just Template:Tmath), and similarly for Template:Tmath.

Convergence is not guaranteed, even for matrices that do have square roots; but if the process converges, the matrix $Y_k$ converges quadratically to a square root Template:Tmath, while $Z_k$ converges to its inverse, Template:Tmath.

By the Babylonian method

Yet another iterative method is obtained by taking the well-known formula of the Babylonian method for computing the square root of a real number, and applying it to matrices. Let Template:Tmath, where Template:Tmath is the identity matrix. The iteration is defined by

Template:Tmath.

Again, convergence is not guaranteed, but if the process converges, the matrix $X_k$ converges quadratically to a square root Template:Tmath. Compared to Denman–Beavers iteration, an advantage of the Babylonian method is that only one matrix inverse need be computed per iteration step. 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.

The Babylonian method follows from Newton's method for the equation $X^2-A=0$ and using $A{X}_k=X_{k}A$ for all Template:Tmath.

Square roots of positive operators

Template:More citations needed 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.Template:Citation needed According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T^1/2 such that T^1/2 is itself positive semidefinite and (T^1/2)² = T. The operator T^1/2 is the unique non-negative square root of T.Template:Citation needed

A bounded non-negative operator on a complex Hilbert space is self-adjoint by definition. So Template:Nobr 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 provides another particular example of square root, which should not be confused with the unique non-negative 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 = A* A = B* B, then there exists a unitary operator U such that A = UB.

Indeed, take B = T^1/2 to be the unique non-negative square root of T. If T is strictly positive, then B is invertible, and so Template:Nobr is unitary:

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

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 Template:Nobr 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

Template:Main 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 Template:Nobr.

If A is not invertible, then it still has a polar decomposition 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

Template:Main By Choi's result, a linear map

$\mathrm{\Phi }:{ℂ}^{n\times n}\to {ℂ}^{m\times m}$

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

$\mathrm{\Phi }(A)=\sum _i^k{V}_{i}A{V}_i^{\ast }$

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

$M_{\mathrm{\Phi }}={\left(\mathrm{\Phi }\left(E_{p,q}\right)\right)}_{p,q}\in {ℂ}^{nm\times nm}$

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

Template:Main 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

$\rho =\sum _i{p}_i{v}_i{v}_i^{\ast }$

where $p_i>0$ and Σ p_i = 1, the set

$\{p_i,v_i\}$

is said to be an ensemble that describes the mixed state ρ. Notice the family {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

$\rho =\sum _j{a}_j{a}_j^{\ast }.$

The trace 1 condition means

$\sum _j{a}_j^{\ast }{a}_j=1.$

Let

$p_i=a_i^{\ast }{a}_i,$

and v_i be the normalized a_i. We see that

$\{p_i,v_i\}$

gives the mixed state ρ.

See also

• Matrix function
• Holomorphic functional calculus
• Logarithm of a matrix
• Sylvester's formula
• Square root of a 2 by 2 matrix

References

• Template:Citation
• Template:Citation, Chapter IV, Reisz functional calculus
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Cite book
• Template:Citation
• Template:Rudin Walter Functional Analysis