The minimum value of the sum of squares of the elements of the matrix

Question

For $D = diag(d_1, d_2, .., d_n)$ a real diagonal $n \times n$ matrix and $X$ a real $n \times n, rank(X) = 1$ matrix define $$F_D(X) = \sum_{i=1}^{n}\sum_{j=1}^{n} (d_{ij} - x_{ij})^{2}.$$ Find the minimum of $F_D(X)$.

Answer 1

It seems you are looking for the best rank-one approximation of a given (diagonal) matrix in the Frobenius norm sense. This is solved by the Eckart-Young Theorem, which tells you that you should use a truncated singular value decomposition, i.e., $X$ is given by $\sigma \cdot u \cdot v^{\rm T}$, where $\sigma$, $u$, and $v$ represent the dominant singular value, left singular vector, and right singular vector, respectively.

In the special case of a diagonal matrix, this solution can be further simplified: as long as the $d_i$ are distinct, the best rank-one approximation is the diagonal matrix that has the largest of the $d_i$'s in the appropriate place and zeros everywhere. The minimum value of $F_d(X)$ is then $\sum_{j\neq i} d_j^2$ where $i = \arg \max_i d_i$.