Uniqueness of the cube root of a symmetric positive definite matrix

Question

After doing many calculus steps I found that $S^3=2I$ where $S\in \mathcal{M}_{m,m}$ is a symmetric positive definite matrix. So I'm wondering if I can write my solution $S\propto I$.

score 3 · Accepted Answer · answered Mar 09 '13 at 00:19

3

The eigenvalues of S all have to be positive real cube roots of 2, so they are all equal. Since any symmetric matrix is diagonalizable, S must be the identity times the real cube root of 2, so S is unique.

answered Mar 09 '13 at 00:19

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Uniqueness of the cube root of a symmetric positive definite matrix

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