# Handling inconsistent solutions obtained by PCA

In order to achieve a 2D representation $X\in\mathbb{R}^{n\times 2}$ of some high-dimensional data residing in $Y\in\mathbb{R}^{n\times k}$, I use PCA:$$X=Y\cdot U,$$where $U\in\mathbb{R}^{k\times 2}$ contains eigenvectors of $Y^TY$ corresponding to its dominant eigenvalues.

However, in case there are multiple occurrences of, e.g., first dominant eigenvalue, my PCA solution, as defined above, will be inconsistent: it would depend on the actual eigenvalue that is declared as 'the first dominant' by the method that I use for the eigendecomposition. What is the recipe to allow for a consistent solution?

Perhaps more important is the following. Namely, PCA provides guarantees on maximal variance along axes; what impact does the above problem have on the solution with maximal variance along axes? Will maximal variance be retained with each solution, regardless on which eigenvector corresponding to dominant first eigenvalue is used?

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 To summarize, the multiplicity of the last eigenvalue considered, $mult(\lambda)$, determines the number of principal components that should be additionally considered, which is $mult(\lambda)-1$. Is this correct? – usero Aug 6 '12 at 6:55 You phrase it in too complicated a way. The number of significant eigenvalues determines the number of components you need to consider. – Wolfgang Bangerth Aug 6 '12 at 7:44 Suppose a PCA solution corresponding to two dominant eigenvalues of the same magnitude is denoted by $X\in\mathbb{R}^{n\times 2}$. Then, besides axis swapping, any rotation of $X$ would also yield a 2D solution with the same variance across $x$ and $y$ axis, subject to axes orthogonality? – usero Nov 16 '12 at 15:14 Yes, that is correct. Because the eigenvalues are the same under rotation. – Wolfgang Bangerth Nov 17 '12 at 11:36