Let $X\in\mathbb{R}^{100\times 100}$ matrix and let its eigenvector and eigenvalues be $X_{vec}$ and $X_{val}$ respectively. If the rank of $X$ is $5$, then is it possible to approximate $X$ with $X_1\in\mathbb{R}^{5\times 5}$?
I have tried to look over internet about dimensionality reduction. Two methods are given: 1. using singular value decomposition(SVD) 2. using principal component analysis(PCA). While the method using SVD does not actually reduce the size of the matrix, it reduces the number of singular values. But this is not what I need. I want, if possible, to reduce the size of the matrix. Similarly, from my understanding, PCA does not reduce the size. So, is there any way to form $X_1$ out of $X$?