Suppose that $M$ is a $3$ by $2$ matrix in which every $2$ by $2$ submatrix is invertible. Is it true that $M$ always has a $2$ by $2$ submatrix $M_{1}$ such that $\left\Vert M_{1}\right\Vert ^{2}\left\Vert M_{1}^{-1}\right\Vert ^{2}\lambda_{2}\left(MM^{T}\right)\ge\left\Vert M\right\Vert ^{2}$ where $\left\Vert \cdot\right\Vert $ is the norm of a matrix (refer to here for the definition) and $\lambda_{2}\left(\cdot\right)$ is the second largest eigenvalue of a $3$ by $3$ symmetric matrix? Thanks
Is this true for $3$ by $2$ matrices?
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linear-algebra
real-analysis
analysis
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0Sounds like this could follow from interlacing inequalities for singular values. You might want to simplify the inequality by expressing it in terms of the singular values of $M_1$ and $M$. – 2012-12-17
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0http://mathoverflow.net/questions/116665/would-this-hold-for-any-2-by-3-matrix – 2012-12-25
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0@peanae: A link is given for the definition of the norm, but the definition linked to is dependent on which norm is given to $\mathbb C^n$, which is not specified above. Are we to presume that $\mathbb C^n$ has the Euclidean norm? – 2013-07-24