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
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
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.