What is Substitution matrices?

A substitution matrix gives $s(a,b)$ for every pair of amino acids as a log-odds score:

where $q_{ab}$ is the frequency with which $a$ and $b$ are found aligned in trusted alignments of related proteins, $p_{a} p_{b}$ the frequency with which they would be paired by chance, and $\lambda$ a scale chosen to make the entries convenient integers. A positive score means the pair occurs more often in homologues than by chance; the identity scores are largest for rare amino acids (tryptophan $+11$, cysteine $+9$ in BLOSUM62) and smallest for common ones (leucine $+4$, alanine $+4$), and conservative substitutions (isoleucine–valine $+3$) score positive while radical ones (tryptophan–glycine $-2$) score negative. The PAM matrices (Dayhoff, 1978) were derived from closely related proteins and extrapolated to greater distances by matrix multiplication; the BLOSUM matrices (Henikoff and Henikoff, 1992) were counted directly in blocks of aligned sequences clustered at a given identity — BLOSUM62 from blocks at $62\,\%$ — and are the default because they were measured, not extrapolated, at the distance where they are used.