The Exercise

Exercise 2.34 asks for the square root and the logarithm of the real $2 \times 2$ matrix $$A = \begin{pmatrix} 4 & 3 \\ 3 & 4 \end{pmatrix}.$$(1)

The matrix $A$ is symmetric, with eigenvalue $7$ on $(|0\rangle + |1\rangle)/\sqrt{2}$ and eigenvalue $1$ on $(|0\rangle - |1\rangle)/\sqrt{2}$. A function $f(A)$ of $A$ is the matrix with the same eigenvectors as $A$ and the eigenvalues $f(7)$ and $f(1)$ on them.

Two Matrices

The library gives both answers in closed form, as real $2 \times 2$ matrices. The square root $\sqrt{A}$ of $A$ is the matrix $$\sqrt{A} = \frac{1}{2} \begin{pmatrix} 1 + \sqrt{7} & \sqrt{7} - 1 \\ \sqrt{7} - 1 & 1 + \sqrt{7} \end{pmatrix}.$$(2)^✦ The logarithm $\log A$ of $A$ is the matrix $$\log A = \frac{\log 7}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix},$$(3)^✦ with $\log 7$ the natural logarithm of $7$.

The Claim

The exponential $e^{M}$ of a square matrix $M$ is the sum of the series $\sum_{k \geq 0} M^k / k!$.

What Is Checked

The claim is stated over the three matrices $A$, $\sqrt{A}$ and $\log A$, written out entry by entry in (1), (2) and (3), and over the exponential series of a matrix.