Worked Problems in Physics

Nielsen and Chuang Exercise 2.34

Posted in Nielsen/Chuang by rpmuller on March 9, 2010

Exercise 2.34 Find the square root and the logarithm of the matrix

\displaystyle  \left[\begin{array}{rr} 4 & 3 \\ 3 & 4 \end{array}\right].

The eigendecomposition of the matrix is:

\displaystyle  \left[\begin{array}{rr} 4 & 3 \\ 3 & 4 \end{array}\right] = \frac{1}{2} \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] \left[\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right] \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]

Thus, the square root and logarithm of the matrix are:

\displaystyle  \sqrt{M} = \frac{1}{2} \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] \left[\begin{array}{rr} 1 & 0 \\ 0 & \sqrt{7} \end{array}\right] \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]

\displaystyle  \log(M) = \frac{1}{2} \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] \left[\begin{array}{rr} 0 & 0 \\ 0 & \log(7) \end{array}\right] \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]