Friday, November 29, 2019

linear algebra - Eigenvalue decomposition of $A = I - xx^T$





Let $A = I - xx^T$, where $x \in \mathbb{R}^n$ and $I$ is the identity matrix of $\mathbb{R}^n$



We know that $A$ is a real symmetric matrix, therefore there exists an eigenvalue decomposition of $A$ such that



$$A = Q^T\Lambda Q$$




Is it possible to find $Q$, $\Lambda$?



$I - xx^T = Q^TQ - xQ^TQx^T = Q^TQ - (Q^Tx)^T(x^TQ)^T...$


Answer



Consider
$$
Ax=(I-xx')x=(1-x'x)x
$$
so $x$ itself is an eigenvector with eigenvalue $1-x'x$. In fact, if $v$ is an eigenvector with some eigenvalue $\alpha$, we have

$$
\alpha v=Av=(I-xx')v=v-(x'v)x\implies(1-\alpha)v=(x'v)x.
$$
This means if $\alpha\neq 1$, then $v$ is proportional to $x$ so in fact $v$ is an eigenvector with eigenvalue $1-x'x$. If $\alpha=1$, then $x'v=0$. Conversely, if $v'x=0$, then $v$ is an eigenvector with eigenvalue $1$:
$$
Av=(I-xx')v=v-(x'v)v=v.
$$
Conclusion: $I-xx'$ has eigenvalues $1-x'x$ and $1$ where $1$ has multiplicity $n-1$. The eigenvectors for $1-x'x$ are parallel to $x$ and the eigenvectors of $1$ are any vector in the space orthogonal to the space spanned by $x$. So you can take $Q'=\Big(\frac{x}{|x|}\;r_1\;\cdots\;r_{n-1}\Big)$ where each $r_i$ is $n\times 1$ and $\{r_1,\ldots,r_{n-1}\}$ is some orthonormal basis of $[\text{span}(x)]^\perp$.


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