The square root of the matrix

The square root of a matrix is an extension of the concept of a numerical square root to a ring of square matrices .

Content

1 Definition
2 Existence and uniqueness
3 Positive definite matrices
4 Literature
5 notes

Definition

Matrix $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ called the square root of the matrix $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ if square $\mathbf {B} ,$ ${\ displaystyle \ mathbf {B},}$ i.e. matrix product $\mathbf {BB} ,$ ${\ displaystyle \ mathbf {BB},}$ matches the matrix $\mathbf {A} .$ ${\ displaystyle \ mathbf {A}.}$

Existence and Uniqueness

Not all matrices have a square root. For example, the matrix has no root $\left({\begin{smallmatrix}0&a\\0&0\end{smallmatrix}}\right),a\neq 0$ ${\ displaystyle \ left ({\ begin {smallmatrix} 0 & a \\ 0 & 0 \ end {smallmatrix}} \ right), a \ neq 0}$ . This matrix is also a zero divisor and the square root of zero. Thus, in a matrix ring, zero has infinitely many square roots.

In cases where the root exists, it is not always uniquely determined. For example, the matrix $\left({\begin{smallmatrix}33&24\\48&57\end{smallmatrix}}\right)$ ${\ displaystyle \ left ({\ begin {smallmatrix} 33 & 24 \\ 48 & 57 \ end {smallmatrix}} \ right)}$ has four roots: $\pm \left({\begin{smallmatrix}1&4\\8&5\end{smallmatrix}}\right)$ ${\ displaystyle \ pm \ left ({\ begin {smallmatrix} 1 & 4 \\ 8 & 5 \ end {smallmatrix}} \ right)}$ and $\pm \left({\begin{smallmatrix}5&2\\4&7\end{smallmatrix}}\right)$ ${\ displaystyle \ pm \ left ({\ begin {smallmatrix} 5 & 2 \\ 4 & 7 \ end {smallmatrix}} \ right)}$ .

Unit matrix ${\bigl (}{\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}{\bigr )}$ ${\ displaystyle {\ bigl (} {\ begin {smallmatrix} 1 & 0 \\ 0 & 1 \ end {smallmatrix}} {\ bigr)}}$ has the following 6 roots among matrices consisting of $0$ ${\ displaystyle 0}$ , $-1$ ${\ displaystyle -1}$ and $+1$ ${\ displaystyle +1}$ :

\left({\begin{matrix}\pm 1&0\\0&\pm 1\end{matrix}}\right),\quad \left({\begin{matrix}\pm 1&0\\0&\mp 1\end{matrix}}\right),\quad \left({\begin{matrix}0&\pm 1\\\pm 1&0\end{matrix}}\right);

{\ displaystyle \ left ({\ begin {matrix} \ pm 1 & 0 \\ 0 & \ pm 1 \ end {matrix}} \ right), \ quad \ left ({\ begin {matrix} \ pm 1 & 0 \\ 0 & \ mp 1 \ end {matrix}} \ right), \ quad \ left ({\ begin {matrix} 0 & \ pm 1 \\\ pm 1 & 0 \ end {matrix}} \ right);}

and also infinitely many symmetric rational square roots of the form:

{\frac {1}{t}}\left({\begin{matrix}s&r\\r&-s\end{matrix}}\right),\quad {\frac {1}{t}}\left({\begin{matrix}s&-r\\-r&-s\end{matrix}}\right),\quad {\frac {1}{t}}\left({\begin{matrix}-s&r\\r&s\end{matrix}}\right),\quad {\frac {1}{t}}\left({\begin{matrix}-s&-r\\-r&s\end{matrix}}\right),

{\ displaystyle {\ frac {1} {t}} \ left ({\ begin {matrix} s & r \\ r & -s \ end {matrix}} \ right), \ quad {\ frac {1} {t}} \ left ({\ begin {matrix} s & -r \\ - r & -s \ end {matrix}} \ right), \ quad {\ frac {1} {t}} \ left ({\ begin {matrix} - s & r \\ r & s \ end {matrix}} \ right), \ quad {\ frac {1} {t}} \ left ({\ begin {matrix} -s & -r \\ - r & s \ end {matrix}} \ right),}

Where $(r,s,t)$ ${\ displaystyle (r, s, t)}$ Is an arbitrary Pythagorean triple , that is, a triple of natural numbers for which $r^{2}+s^{2}=t^{2}$ ${\ displaystyle r ^ {2} + s ^ {2} = t ^ {2}}$ .

The difficulty of extracting the root from the matrix is due to the fact that the matrix ring is non-commutative and has zero divisors, that is, it is not a domain of integrity . In the field of integrity, for example, in the ring of polynomials above a field , every element has no more than two square roots.

Positive definite matrices

A positive definite matrix always has exactly one positive definite root, which is called an arithmetic square root ^[1] .

All in all, a positive definite matrix $A$ ${\ displaystyle A}$ of order $n$ ${\ displaystyle n}$ with different eigenvalues has $2^{n}$ ${\ displaystyle 2 ^ {n}}$ roots. Expanding such a matrix in eigenvectors, we obtain its representation in the form $\mathbf {A} =\mathbf {VDV^{-1}} ,$ ${\ displaystyle \ mathbf {A} = \ mathbf {VDV ^ {- 1}},}$ Where $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ - diagonal matrix with eigenvalues $\lambda _{i}>0$ ${\ displaystyle \ lambda _ {i}> 0}$ . Then the square roots of the matrix $A$ ${\ displaystyle A}$ have the form $\mathbf {VD^{\frac {1}{2}}V^{-1}} ,$ ${\ displaystyle \ mathbf {VD ^ {\ frac {1} {2}} V ^ {- 1}},}$ Where $\mathbf {D^{\frac {1}{2}}}$ ${\ displaystyle \ mathbf {D ^ {\ frac {1} {2}}}}$ - diagonal matrix with elements $\pm {\sqrt {\lambda _{i}}}$ ${\ displaystyle \ pm {\ sqrt {\ lambda _ {i}}}}$ on the diagonal.

Literature

Gantmakher F.R. Matrix theory. M .: GITTL, 1953, S. 212-219.
Voevodin V.V., Voevodin Vl. B. Encyclopedia of linear algebra. Electronic system LINEAL. SPb .: BHV-Petersburg, 2006.

Notes

↑ Valentin Vasilievich Voevodin, Yuri Alekseevich Kuznetsov. Matrices and calculations . - "Science," Chap. ed. physical and mathematical literature, 1984. - S. 88-89. - 330 p.

[1] Valentin Vasilievich Voevodin, Yuri Alekseevich Kuznetsov. Matrices and calculations . - "Science," Chap. ed. physical and mathematical literature, 1984. - S. 88-89. - 330 p.