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I am working on a problem.

For a $4$x$4$ matrix M and an arbitrary vector $y$ of length $4$ such that $y^Ty=1$, I know the output $\lambda$ of the matrix multiplication $y^TMy = \lambda$. I also know the vector $y$. Given the matrix $M$ is symmetric and positive semi-definite, are there ways to reconstruct the matrix $M$? How?

lakdee
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    if $M$ is symmetric, it is easy and definiteness is irrelevant. The four diagonal values are found with $y = e_i ; . ;$ Next for $i \neq j,$ take $y = e_i + e_j$ – Will Jagy May 25 '20 at 02:38
  • You can think of $y^TMy$ as a quadratic form, the corresponding symmetric bilinear form $x^TMy$ can be recovered via the polarization identity. Substituting standard basis vectors into the bilinear form generated by $M$ directly gives you its entries. – Conifold May 25 '20 at 03:37

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No, there isn't not enough information to reconstruct $M$, but we can write down the general solution to your problem. I suppose that $\lambda\ge0$, otherwise $M$ cannot be positive semidefinite. Pick a fixed orthogonal matrix $Q$ whose first column is $y$. In the below, $S$ denotes an an arbitrary $3\times3$ positive semidefinite matrix and $x$ denotes an arbitrary vector in $\mathbb R^3$.

  • When $\lambda=0$, the general solution is given by $$ M=Q\pmatrix{0&0\\ 0&S}Q^T. $$
  • When $\lambda>0$, the general solution is given by $$ M=Q\pmatrix{\lambda&x^T\\ x&\frac{1}{\lambda}xx^T+S}Q^T. $$
user1551
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