Invariance Properties and Evaluation Metrics Derived from the Confusion Matrix in Multiclass Classification

Multiclass evaluation metrics derived from the confusion matrix—such as accuracy, precision, and recall—are widely used yet rarely formalized with respect to their structural invariance. This paper introduces a closed-form analytical framework demonstrating that core metrics remain stable under confusion matrix transposition and class label permutation. A minimal sufficient set of metrics is identified—Overall accuracy, Macro-averaged precision, Macro-averaged recall, and Weighted precision—from which all other commonly reported metrics can be derived. These invariance properties are established algebraically and validated numerically using both the Wine dataset and CIFAR-10 with a convolutional neural network. The results confirm robustness across models and datasets and clarify metric behavior under structural transformations. The proposed framework reinforces reproducibility and interpretability in multiclass evaluation, with implications for diagnostic screening, fraud detection, and classification audits.