give its nth power. If Q is diagonalizable, the matrix exponential can be computed directly: let Q = U−1 Λ U be a diagonalization of Q, with where Λ is a diagonal matrix and where are the eigenvalues of Q, each repeated according to its multiplicity. Then where the diagonal matrix eΛt is given by Generalised time reversible Generalised time reversible (GTR) is the most general neutral, independent, finite-sites, time-reversible model possible. It was first described in a general form by Simon Tavaré in 1986. The GTR model is often called the general time reversible model in publications; it has also been called the REV model. The GTR parameters for nucleotides consist of an equilibrium base frequency vector, , giving the frequency at which each base occurs at each site, and the rate matrix Because the model must be time reversible and must approach the equilibrium nucleotide (base) frequencies at long times, each rate below the diagonal equals the reciprocal rate above the diagonal multiplied by the equilibrium ratio of the two bases. As such, the nucleotide GTR requires 6 substitution rate parameters and 4 equilibrium base frequency parameters. Since the 4 frequency parameters must sum to 1, there are only 3 free frequency parameters. The total of 9 free parameters is often further reduced to 8 parameters plus , the overall number of substitutions per unit time. When measuring time in substitutions (=1) only 8 free parameters remain. In general, to compute the number of parameters, you count the number of entries above the diagonal in the matrix, i.e. for n trait values per site , and then add n-1 for the equilibrium frequencies, and subtract 1 because is fixed. You get For example, for an amino acid sequence (there are 20 "standard" amino acids that make up proteins), you would find there are 208 parameters. However, when studying coding regions of the genome, it is more common to work with a codon substitution model (a codon is three bases and codes for one amino acid in a protein). There are codons, resulting in 2078 free parameters. However, the rates for transitions between codons which differ by more than one base are often assumed to be zero, reducing the number of free parameters to only parameters. Another common practice is to reduce the number of codons by forbidding the stop (or nonsense) codons. This is a biologically reasonable assumption