# Eigenvectors normalization

When eigenvectors are real, they are normalized to 1. How are complex eigenvectors normalized?

If there is at least one floating point number (real or complex) somewhere in the matrix A: Yes, vectors are normalized (with respect to the standard Hermitian inner product #1.Conjugate[#2] &). They are also orthogonalized if eigenspaces of A are known to be orthogonal, e.g. when a represents a normal operator.
• I don't get what you mean. The norm induced by the Hermitian inner product #1.Conjugate[#2] & is Sqrt[Abs[Conjugate[#].#]] &. This is identical to Norm for complex vectors. And floating point eigenvectors are normalized with respect to precisely this norm. – Henrik Schumacher Aug 5 '18 at 22:01
• In order to make everything clear, here is example a = {{2. + 3. I, 1. - 2. I, 4.}, {4. - 3. I, 3. + 1.5 I, -2. I}, {5. + 3. I, 6. + 4. I, 1.7 - 2. I}}; b = Eigenvectors[a] {{0.397675\[VeryThinSpace]+ 0.0415786 I, 0.397957\[VeryThinSpace]- 0.304432 I, 0.767514\[VeryThinSpace]+ 0. I}, {0.701712\[VeryThinSpace]+ 0. I, -0.663209 - 0.157879 I, -0.202508 + 0.0426597 I}, {-0.30459 + 0.404155 I, -0.0862046 - 0.116806 I, 0.850181\[VeryThinSpace]+ 0. I}}  Please show me on this example what normalization took place. – user1765636 Aug 6 '18 at 21:05
• Norm /@ b returns {1., 1., 1.}. Note that eigenvectors are returned as rows not as columns. – Henrik Schumacher Aug 6 '18 at 21:14