Suppose that:

(*) for every vector

.

Let

i,

j be any two particular coordinate indices.

(i.e. these are NOT abstract indices, they're constants, in the range 0..3 when working in four dimensions).

Define the vectors:

p =

q =

r =

(e.g. for

i=1 and

j=3,

p = [0,1,0,0],

q = [0,0,0,1],

r = [0,1,0,1]; for

i=

j=0,

p,

q = [1,0,0,0],

r = [2,0,0,0]; ...etc.)

Then:

.

Combining the above gives:

(1) And similarly:

(2) By

(*), the RHS of

(1) and of

(2) must be equal; so we have proven that

for every choice of

i,

j.

Hence

.

If

and

are symmetric tensors,

and

, and so in that case

(*) implies that

.