Are there many ways to decompose a matrix, ${\bf{A}} \in \mathbb{R}^{n \times n}$, into a sum of rank-$1$ matrices?
I ask because I know that if a matrix is symmetric and diagonalizable, $\bf{A}=\Phi\Lambda\Phi^{T}$, then you can use the eigendecomposition to form a sum of rank-$1$ matrices:
$$\bf{A} = \sum_{i=1}^{n}\lambda_i \phi_i\phi_i^T\,.$$
It is also true that you can always use the singular value decomposition (SVD) to do this, as well:
$$\bf{A} = \sum_{i=1}^n \sigma_i \bf{u}_i\bf{v}_i^T\,.$$
So, unless these happen to be equal, which is not true in general, then I have found two ways already. Are there infinitely many ways? What is an example of another decomposition of this form?
Best Answer
Yes, there exists an infinity of such decompositions as a sum of rank-1 matrices.
It suffices to use any factorisation of $A$ as a product of 2 matrices:
$$A=MN=\sum_{i=1}^n C_i L_i \tag{1}$$
where $C_i$ is the $i$th column of matrix $M$ and $L_i$ is the $i$th line of matrix $N$
Why are there an infinity of ways to write (1) ? It suffices to take any invertible matrix $M$, and to associate it $N:=M^{-1}A$.
Remark : in order to understand (1), let us consider the $2 \times 2$ case :
$$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}e&f\\g&h\end{pmatrix}=\begin{pmatrix}a\\ c\end{pmatrix}\begin{pmatrix}e&f\end{pmatrix}+\begin{pmatrix}b\\d\end{pmatrix}\begin{pmatrix}g&h\end{pmatrix}$$