# Square rooting of a diagonalizable matrix

Square root of a diagonalizable matrix $$A$$ can be otained as $$B=\sqrt{A}=D \Lambda D^{-1} ~~~~(1),$$ where $$\Lambda=\begin{pmatrix} \lambda_1^{1/2} & 0 \\ 0 & \lambda_2^{1/2} \end{pmatrix}.$$ Here, $$\lambda_{1,2}$$ are the eigenvalues and $$D$$ is the diagonalizing matrix od $$A$$. Recently, in a question:

Solving a matrix equation of four unknowns

Square root of $$P=\begin{pmatrix} 4 & 5 \\ 3 & 6 \end{pmatrix},$$ was required such that $$P=Q^2$$. The eigenvalues of $$P$$ are $$9,1$$, the method (1) and also the the command `MatrixPower[P, 1/2]' of Mathematica gives $$Q=\frac{1}{4}\begin{pmatrix} 7 & 5 \\ 3 & 9 \end{pmatrix}.~~~~(2)$$ However a brute force method of construction $$Q$$ from four unknowns $$a,b,c,d$$ and solutions of the equations thereby along with the conditions that $$Trace (P)=Trace (Q^2)$$ and $$\det(P)=\det(Q^2)$$ yields a matrix $$Q= \frac{1}{2} \begin{pmatrix} 1 & 5 \\ 3 & 3 \end{pmatrix}.~~~~(3)$$ Can one explain why does the method in (1) and the Mathematica igonres (3)? How to resolve this ?

• The matrix square root isn't unique, as your example clearly illustrates (it is true that a symmetric and positive definite matrix has a unique symmetric and positive definite square root, but that breaks when the matrices aren't symmetric.) What is your actual question here? – Brian Borchers Jun 22 '19 at 3:56
• @Brain Borchers, Thanks, the question is how does one get (3) from the method (1).? – Z Ahmed Jun 22 '19 at 4:08
• To underscore @BrianBorchers comment, how many matrices $A$ are there such that $A^2=I$? – amd Jun 22 '19 at 4:39
• @BrianBrochers You may also see dan_filea's solution. – Z Ahmed Jun 22 '19 at 6:50

Let us diagonalize explicitly. $$\begin{bmatrix} 4&5\\3&6 \end{bmatrix} = \begin{bmatrix} 5 & 5 \\ 5 & -3 \end{bmatrix} \begin{bmatrix} 9&\\&1 \end{bmatrix} \begin{bmatrix} 5 & 5 \\ 5 & -3 \end{bmatrix}^{-1} \ ,$$ A square root of a slightly more general shape is then: $$Q= \begin{bmatrix} 5 & 5 \\ 5 & -3 \end{bmatrix} \begin{bmatrix} \pm 3&\\&\pm 1 \end{bmatrix} \begin{bmatrix} 5 & 5 \\ 5 & -3 \end{bmatrix} ^{-1} \ .$$ Then the four matrices obtained in this way are: $$\left[\begin{array}{rr} -\frac{7}{4} & -\frac{5}{4} \\ -\frac{3}{4} & -\frac{9}{4} \end{array}\right] \ ,\ \left[\begin{array}{rr} -\frac{1}{2} & -\frac{5}{2} \\ -\frac{3}{2} & -\frac{3}{2} \end{array}\right] \ ,\ \left[\begin{array}{rr} \frac{1}{2} & \frac{5}{2} \\ \frac{3}{2} & \frac{3}{2} \end{array}\right] \ ,\ \left[\begin{array}{rr} \frac{7}{4} & \frac{5}{4} \\ \frac{3}{4} & \frac{9}{4} \end{array}\right] \ .$$