It is proved that a necessary and sufficient condition for a general design to be balanced is that the matrix of the adjusted normal equations for the estimates of treatment effects has $v - 1$ equal latent roots other than zero.
"A Note on Balanced Designs." Ann. Math. Statist. 29 (1) 290 - 294, March, 1958. https://doi.org/10.1214/aoms/1177706729