= Solution
The Dirichlet difference <Laplacian> has an orthogonal sine basis with <eigenvalues> $-\eta_j/h^2$, where $\eta_j=4\sin^2(j\pi/(2M))$ for $j=1,\ldots,M-1$. On a mode the <theta method> amplification is
$$
G(s)=\frac{1-(1-a)s}{1+as},\qquad s=\mu\eta_j>0.
$$
If $a\geq1/2$, the denominator is positive and
$$
(1+as)^2-[1-(1-a)s]^2=2s+(2a-1)s^2\geq0.
$$
Thus $|G(s)|\leq1$ for every mode and every $\mu>0$. Orthogonality gives a uniform discrete <L2 norm> contraction, not merely individual eigenvalue estimates.
If $a<0$, a positive $s=-1/a$ makes the implicit matrix singular. If $a=0$, $G=1-s$ becomes arbitrarily large. If $0<a<1/2$, $G(s)$ tends to $-(1-a)/a$, whose magnitude is greater than one. Hence \b[unconditional stability holds exactly for $a\geq1/2$].
For completeness, $a<1/2$ is contractive under $\mu\eta_{\max}\leq2/(1-2a)$; the mesh-independent sufficient restriction is $\mu\leq1/[2(1-2a)]$. This follows from the same squared-modulus difference and keeps the denominator positive even for negative $a$. At $a=1/2$, very high frequencies approach amplification $-1$, so <A-stability> does not imply damping of rough heat data. At $a=1$ the <Backward Euler method> damps them strongly; this is the <rough-data damping distinction for the theta method>.
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