The characteristic roots of the information matrix $(M)_T$ of a balanced $2^m$ fractional factorial design $T$ are obtained, when the parameters to be estimated include the general mean $\mu$, the main effect $A_i$, and the two-factor interaction $A_iA_j$ (briefly, $A_{ij}$), the remaining effects being assumed negligible. (If $(M)_T$ is nonsingular, $T$ is a design of resolution $V$.) It is well known that $T$ depends on five nonnegative integers $(\mu_0, \mu_1, \mu_2, \mu_3, \mu_4)$, called its "index set." In Srivastava (1970), the special case when $\mu_0 = \mu_4$ and $\mu_1 = \mu_3$ was considered; in this paper, the theory is presented for the general case. As a by-product of this work, we obtain a class of useful necessary conditions on the set $(\mu_0, \mu_1, \mu_2, \mu_3, \mu_4)$ such that a design $T$ with this index set may (combinatorially) exist. If $(M)_T$ is nonsingular, and $(V)_T = \lbrack(M)_T\rbrack^{-1}$, an explicit expression (as a function of the $\mu_i$) has been obtained for $\operatorname{tr}(V)_T$; similar expressions for $|(V)_T|$ and $\operatorname{ch}_{\max}(V)_T$ can be easily written down using our results. One reason why $\operatorname{tr}(V)_T$ (rather than the other two criteria) should be used for comparing balanced resolution $V$ fractions is given. Finally, it is shown (through an example of a previously unknown design with resolution $V m = 7$) that for a given $N$ (the number of runs), an (existing) optimal balanced design (optimal with respect to, say, the trace criterion) does not necessarily satisfy the restriction $(\mu_0 = \mu_4$ and $\mu_1 = \mu_3$), and may be distinct from the design which is optimal in the restricted class. (Scores of other such examples may be found in Srivastava and Chopra (1970a), where the results of this paper are used in a basic manner.) Thus the need for considering designs with general index sets (which is accomplished in the present paper) becomes obvious.