Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
More tensor (matrix) algebra?
1. if T_ij = −T_ji and S_ij = S_ji, then T_ijS_ij = 0.
2. Show that if epsilon_ijkT_jk = 0 then T_ij = T_ji.
1 Answer
- EugeneLv 77 years agoFavorite Answer
1. Since T_ij = -T_ji and S_ij = S_ji,
T_ij S_ij = (-T_ji) S_ji = -T_ji S_ji = -T_ij S_ij,
and thus T_ij S_ij = 0.
2. I'll use "e" for epsilon and "d" for delta. Fix indices a and b. By hypothesis, it follows that
(1) e_iab e_ijk T_jk = 0.
By the epsilon-delta identity,
(2) e_iab e_ijk = d_aj d_bk - d_ak d_bj.
Combining (1) and (2) results in
(3) (d_aj d_bk - d_ak d_bj) T_jk = 0
that is,
T_ab - T_ba = 0.
Therefore T_ab = T_ba.
