Square root of a 2 by 2 matrix

A square root of a 2×2 matrix Template:Mvar is another 2×2 matrix Template:Mvar such that Template:Math, where Template:Math stands for the matrix product of Template:Mvar with itself. In general, there can be zero, two, four, or even an infinitude of square-root matrices. In many cases, such a matrix Template:Mvar can be obtained by an explicit formula.

Square roots that are not the all-zeros matrix come in pairs: if Template:Mvar is a square root of Template:Mvar, then Template:Math is also a square root of Template:Mvar, since Template:Math.

A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root.

A general formula

The following is a general formula that applies to almost any 2 × 2 matrix.Template:R Let the given matrix be $$M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right),$$ where Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar may be real or complex numbers. Furthermore, let Template:Math be the trace of Template:Mvar, and Template:Math be its determinant. Let Template:Mvar be such that Template:Math, and Template:Mvar be such that Template:Math. That is, $$s=\pm \sqrt{\delta },t=\pm \sqrt{\tau +2s}.$$ Then, if Template:Math, a square root of Template:Mvar is $$R=\frac{1}{t}\left(\begin{array}{cc}A+s& B\\ C& D+s\end{array}\right)=\frac{1}{t}(M+sI).$$

Indeed, the square of Template:Mvar is $$\begin{array}{cc}{R}^2& =\frac{1}{t^2}\left(\begin{array}{cc}{A}^2+BC+2sA+s^2& AB+BD+2sB\\ CA+DC+2sC& CB+D^2+2sD+s^2\end{array}\right)\\ & =\frac{1}{t^2}\left(\begin{array}{cc}{A}^2+BC+2sA+AD-BC& AB+BD+2sB\\ AC+CD+2sC& BC+D^2+2sD+AD-BC\end{array}\right)\\ & =\frac{1}{A+D+2s}\left(\begin{array}{cc}A(A+D+2s)& B(A+D+2s)\\ C(A+D+2s)& D(A+D+2s)\end{array}\right)=M.\end{array}$$

Note that Template:Mvar may have complex entries even if Template:Mvar is a real matrix; this will be the case, in particular, if the determinant Template:Mvar is negative.

The general case of this formula is when Template:Mvar is nonzero, and Template:Math, in which case Template:Mvar is nonzero, and Template:Mvar is nonzero for each choice of sign of Template:Mvar. Then the formula above will provide four distinct square roots Template:Mvar, one for each choice of signs for Template:Mvar and Template:Mvar.

Special cases of the formula

If the determinant Template:Mvar is zero, but the trace Template:Mvar is nonzero, the general formula above will give only two distinct solutions, corresponding to the two signs of Template:Mvar. Namely, $$R=\pm \frac{1}{t}\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)=\pm \frac{1}{t}M,$$ where Template:Mvar is any square root of the trace Template:Mvar.

The formula also gives only two distinct solutions if Template:Mvar is nonzero, and Template:Math (the case of duplicate eigenvalues), in which case one of the choices for Template:Mvar will make the denominator Template:Mvar be zero. In that case, the two roots are $$R=\pm \frac{1}{t}\left(\begin{array}{cc}A+s& B\\ C& D+s\end{array}\right)=\pm \frac{1}{t}(M+sI),$$ where Template:Mvar is the square root of Template:Mvar that makes Template:Math nonzero, and Template:Mvar is any square root of Template:Math.

The formula above fails completely if Template:Mvar and Template:Mvar are both zero; that is, if Template:Math, and Template:Math, so that both the trace and the determinant of the matrix are zero. In this case, if Template:Mvar is the null matrix (with Template:Math), then the null matrix is also a square root of Template:Mvar, as is any matrix $$R=\left(\begin{array}{cc}0& 0\\ c& 0\end{array}\right)\text{and}R=\left(\begin{array}{cc}0& b\\ 0& 0\end{array}\right),$$ where Template:Mvar and Template:Mvar are arbitrary real or complex values. Otherwise Template:Mvar has no square root.

Formulas for special matrices

Idempotent matrix

If Template:Mvar is an idempotent matrix, meaning that Template:Math, then if it is not the identity matrix, its determinant is zero, and its trace equals its rank, which (excluding the zero matrix) is Template:Math. Then the above formula has Template:Math and Template:Math, giving Template:Mvar and Template:Math as two square roots of Template:Mvar.

Exponential matrix

If the matrix Template:Mvar can be expressed as real multiple of the exponent of some matrix Template:Mvar, $M=r\exp (A)$, then two of its square roots are $\pm \sqrt{r}\exp \left(\frac{1}{2}A\right)$. In this case the square root is real.Template:R

Diagonal matrix

If Template:Mvar is diagonal (that is, Template:Math), one can use the simplified formula $$R=\left(\begin{array}{cc}a& 0\\ 0& d\end{array}\right),$$ where $a=\pm \sqrt{A}$, and $d=\pm \sqrt{D}$. This, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both Template:Mvar and Template:Mvar are zero, respectively.

Identity matrix

Because it has duplicate eigenvalues, the 2×2 identity matrix $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ has infinitely many symmetric rational square roots given by $$\frac{1}{t}\left(\begin{array}{cc}s& r\\ r& -s\end{array}\right)\text{ and }\left(\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right),$$ where Template:Math are any complex numbers such that $r^2+s^2=t^2.$Template:R

Matrix with one off-diagonal zero

If Template:Mvar is zero, but Template:Mvar and Template:Mvar are not both zero, one can use $$R=\left(\begin{array}{cc}a& 0\\ \frac{C}{a+d}& d\end{array}\right).$$

This formula will provide two solutions if Template:Math or Template:Math or Template:Math, and four otherwise. A similar formula can be used when Template:Mvar is zero, but Template:Mvar and Template:Mvar are not both zero.

Real matrices with real square roots

The algebra Template:Math of 2×2 real matrices has three types of planar subalgebras. Each subalgebra admits the exponential map. If $p=\exp (q)$, then $\pm \exp \left(\frac{q}{2}\right)$ are square roots of Template:Mvar. The condition that the matrix is the image under exp limits it to half the plane of dual numbers, and to a quarter of the plane of split complex numbers, but does not constrain ordinary complex planes since the exponential mapping covers them. In the split-complex case there are two more square roots of Template:Mvar since each quadrant contains one.