Let $Y\left(t\right)$ be a matrix function, then does $\frac{d}{dt}Y^{T}Y=O$ necessarily imply $Y^{T}\left(t\right)Y\left(t\right)\equiv I$ ? Why or why not? Here $Y^{T}$ the transpose of $Y$ , $O$ zero matrix and $I$ the identity matrix.
Differential equation for a matrix-valued function
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matrices
derivatives
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0What did you learn? What have you tried? – 2012-02-29
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0Have you tried a 2 by 2 case to see what may happen? – 2012-02-29
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2Do you have any kind of initial condition? Otherwise the answer is trivially no: If $X$ is any matrix with the property that $X^tX\neq I$, then setting $Y(t) = X$ (constant matrix) gives a "no" answer. – 2012-02-29
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0@Jack Let's exclude the trivial case when $Y\left(t\right)=$const. At first sight, it appears that $Y^{T}Y$ can be any constant matrix, but what I am hoping is that somebody can give me a proof that this is not the case, that the only choice is the identity matrix. I konw there is something special about $Y^{T}Y$, such as symmetric, so definitely not any constant matrix ! – 2012-02-29