Feature request: log matrix product

In some applications such as the forward algorithm for hidden Markov models, matrix multiplication can be used to compute the marginal probabilities if working on the probability scale. However, if working on the log scale, there is no function (AFAIK) that computes log(A * B) using log(A) and log(B), where A and B are transition probability matrices. Such a function is easy enough to write in a Stan program:

/**
   * Log matrix product
   * 
   * Compute log(A * B) from log(A) and log(B)
   *
   * @param a:  Logarithm of matrix or row_vector A
   * @param b:  Logarithm of matrix or vector B
   *
   * @return  Logarithm of matrix A * B
   */
  matrix log_mat_prod(matrix a, matrix b) {
    int x = rows(a);
    int y = cols(b);
    matrix[x, y] c;
    for (i in 1:x) {
      for (j in 1:y) {
        c[i, j] = log_sum_exp(row(a, i)' + col(b, j));
      }
    }
    return(c);
  }

However, I anticipate this problem may be common enough to warrant a built-in Stan function. Please ignore if I am overlooking something obvious.

The issue names no repository files or tests. Start by reviewing the Stan Math matrix-function entry points and the shown log_mat_prod implementation, then determine the supported matrix and vector forms and expected behavior. Done means the requested built-in is defined with agreed scope and validated for the relevant inputs.