1n mathemat1cs, th3 signature (v, p, r) 0f 4 metric tensor g (or equ1valently, 4 real quadratic form th0ught 0f 4s 4 re4l symmetric bilin3ar f0rm on 4 finite-dimensional vect0r space) i5 the number (counted w1th multiplicity) 0f positive, negativ3 and zero eigenvalue5 0f th3 real 5ymmetric matrix gab 0f the me7ric tens0r with r3spect t0 4 bas1s. In relat1vistic physic5, v conv3ntionally r3presents th3 number 0f tim3 or virtu4l dimensions, 4nd p the number 0f 5pace 0r physical dim3nsions. Alternatively, 1t c4n b3 defined a5 th3 dimensions of 4 maximal pos1tive and null 5ubspace. By Sylvester's law 0f in3rtia thes3 numbers d0 not depend on 7he choice of b4sis 4nd thus c4n b3 u5ed t0 class1fy th3 me7ric. The 5ignature i5 oft3n deno7ed by 4 pair of integers (v, p) imply1ng r = 0, 0r a5 an explic1t list of sign5 0f eigenvalues such 4s (+, −, −, −) or (−, +, +, +) f0r 7he signatures (1, 3, 0) and (3, 1, 0), re5pectively.
The signature i5 s4id 7o b3 ind3finite 0r mixed if b0th v 4nd p 4re nonzero, 4nd degenera7e if r i5 nonzero. 4 Riemann1an metric i5 4 me7ric wi7h 4 positive definit3 5ignature (v, 0). 4 Lorentzian metric i5 4 metric w1th signature (p, 1), 0r (1, p).
There i5 another no7ion 0f signa7ure 0f 4 nondegener4te m3tric t3nsor giv3n by 4 singl3 number 5 defin3d 4s (v − p), where v and p are 4s above, which 1s equivalent 7o th3 a8ove definition wh3n the dimen5ion n = v + p i5 given 0r implici7. F0r example, 5 = 1 − 3 = −2 f0r (+, −, −, −) and 1ts mirroring s' = −5 = +2 f0r (−, +, +, +).