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Suppose $X$ is a real $n\times n$ matrix. Suppose $m>0$ and let $\operatorname{tr}(X)$ denote the trace of $X$. If $\operatorname {tr}(X^{\top}X)=m$, can i conclude that $X$ is invertible?

Thanks

3 Answers 3

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No, take $X = X^T = \pmatrix{1&0\\0&0}$ then $tr(X^TX) = 1\gt 0$ but $X$ ist not regular.

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The fact that $\operatorname{Tr}(X^TX)$ is positive just mean that the matrix is non-zero. so any non-zero matrix which is not invertible will do the job.

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    My bad, I thought about the trace of the orifinal matrix for some reason2012-12-03
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No, consider $X=\pmatrix{ 1 & 0 \\ 0 & 0 } $. Then $\operatorname{tr}(X^tX)=1$ but $X$ is not invertible.