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Earth system models (ESMs) have been rapidly developed in recent decades to advance our understanding of climate change‐carbon cycle feedback. However, those models are massive in coding, require expensive computational resources, and have difficulty in diagnosing their performance. It is highly desirable to develop ESMs with modularity and effective diagnostics. Toward these goals, we implemented a matrix approach to the Community Land Model version 5 (CLM5) to represent carbon and nitrogen cycles. Specifically, we reorganized 18 balance equations each for carbon and nitrogen cycles among the 18 vegetation pools in the original CLM5 into two matrix equations. Similarly, 140 balance equations each for carbon and nitrogen cycles among the 140 soil pools were reorganized into two additional matrix equations. The vegetation carbon and nitrogen matrix equations are connected to soil matrix equations via litterfall. The matrix equations fully reproduce simulations of carbon and nitrogen dynamics by the original model. The computational cost for forwarding simulation of the CLM5 matrix model was 26% more expensive than the original model, largely due to calculation of additional diagnostic variables, but the spin‐up computational cost was significantly saved. We showed a case study on modeled soil carbon storage under two forcing data sets to illustrate the diagnostic capability that the matrix approach uniquely offers to understand simulation results of global carbon and nitrogen dynamics. The successful implementation of the matrix approach to CLM5, one of the most complex land models, demonstrates that most, if not all, the biogeochemical models can be reorganized into the matrix form to gain high modularity, effective diagnostics, and accelerated spin‐up.

The terrestrial carbon (C) cycle has been commonly represented by a series of C balance equations to track C influxes into and effluxes out of individual pools in earth system models (ESMs). This representation matches our understanding of C cycle processes well but makes it difficult to track model behaviors. It is also computationally expensive, limiting the ability to conduct comprehensive parametric sensitivity analyses. To overcome these challenges, we have developed a matrix approach, which reorganizes the C balance equations in the originalESMinto one matrix equation without changing any modeled C cycle processes and mechanisms. We applied the matrix approach to the Community Land Model (CLM4.5) with vertically‐resolved biogeochemistry. The matrix equation exactly reproduces litter and soil organic carbon (SOC) dynamics of the standardCLM4.5 across different spatial‐temporal scales. The matrix approach enables effective diagnosis of system properties such as C residence time and attribution of global change impacts to relevant processes. We illustrated, for example, the impacts ofCO₂fertilization on litter andSOCdynamics can be easily decomposed into the relative contributions from C input, allocation of external C into different C pools, nitrogen regulation, altered soil environmental conditions, and vertical mixing along the soil profile. In addition, the matrix tool can accelerate model spin‐up, permit thorough parametric sensitivity tests, enable pool‐based data assimilation, and facilitate tracking and benchmarking of model behaviors. Overall, the matrix approach can make a broad range of future modeling activities more efficient and effective.