i^T\mathbf{X}_i\mathbf{e}},$$ and notice that $\mathbf{b}_j^T\mathbf{e}=0$ (because $\mathbf{b}_j$ and $\mathbf{e}$ are eigenvectors corresponding to different eigenvalues of $\mathbf{B}$) to produce $$\mathbf{B}_i=\mathbf{B}-\sum_{j=1}^{i-1}\beta_j\mathbf{b}_j\mathbf{b}_j^T=\mathbf{B}_{i-1}-\beta_{i-1}\mathbf{b}_{i-1}\mathbf{b}_{i-1}^T.$$ So by Brauer’s theorem again, the eigenpairs of $\mathbf{B}_i$ are the ones of $\mathbf{B}_{i-1}$ with $(\beta_{i-1}, \mathbf{b}_{i-1})$ replaced by an eigenpair with zero eigenvalue. So $\beta_i$ is the largest eigenvalue of $\mathbf{B}_i$ and $\mathbf{b}_i$ is the eigenvector of $\mathbf{B}_i$ corresponding to $\beta_i$.

Theorem \[thm7\] says when we build the new data matrix $\mathbf{X}_i$ from $\mathbf{X}$, $\mathbf{d}_i$ and $m_i$ change. Also $\mathbf{B}_i$ is different from $\mathbf{B}$, but the eigenpairs of $\mathbf{B}$ are retained by $\mathbf{B}_i$ except for the first $i-1$ pairs. The conclusion is that the first modularity component has the largest modularity of the data $\mathbf{X}$. Each succeeding modularity component has the largest modularity with the constraint that it is orthogonal to all previous modularity components.

Conclusion
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In this paper, the concept of modularity components is defined, and some important properties of modularity components are proven. The concept of modularity components can be used to explain why using more than one eigenvectors of the modularity matrix to do data clustering is reasonable. The combination of modularity clustering and modularity components gives a modularity component analysis that has some nice properties similar to the well known principal component analysis.

Appendices {#appendices .unnumbered}
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Proof of Lemma \[thm3\] {#app1}
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The lemma is based on a theorem from [@bunch1978rank] about the interlacing property of a diagonal matrix and its rank-one modification and how to calculate the eigenvectors of a diagonal plus rank one (DPR1) matrix [@meyer2000matrix]. The