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How to embed a matrix, for example, a $3x3$ singular matrix in to $R^9$?

How to compute the induced metric? Is it just the Frobenius norm of the matrix?

Many Thanks. sam

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    The question is unclear. For instance -- and this is just for starters; there are more serious objections -- "a matrix" is just a single object, so to embed "a matrix" into $\mathbb{R}^{n^2}$ is equivalent to just picking a point in $\mathbb{R}^{n^2}$. Doubtless this is not what you really mean...2011-01-02

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What restrictions are there on the embedding? If you just mean a bijection between $3 \times 3$ matrices and points in $\mathbb{R^9}$, there are many. As there are $9$ entries in a $3 \times 3$ matrix, there are easy bijections taking each entry in the matrix to one component. The bijection allows an easy metric, too-just use the usual metric in $\mathbb{R^9}$ It is true that the singular matrices have a constraint that reduces them to an eight-dimensional space. This is the source of the first question-this embedding ignores the impact of singularity.

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    i will get back on this.2011-01-02