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complex conjugate∗
akrowne†
2013-03-21 13:14:10
1
Definition
1.1
Scalar Complex Conjugate
Let z be a complex number with real part a and imaginary part b,
z = a + bi
Then the complex conjugate of z is
z̄ = a − bi
Complex conjugation represents a reflection about the real axis on the Argand diagram representing a complex number.
Sometimes a star (∗) is used instead of an overline, e.g. in physics you might
see
Z ∞
Ψ∗ Ψdx = 1
−∞
∗
where Ψ is the complex conjugate of a wave function.
1.2
Matrix Complex Conjugate
Let A = (aij ) be a n × m matrix with complex entries. Then the complex
conjugate of A is the matrix A = (aij ). In particular, if v = (v 1 , . . . , v n ) is a
complex row/column vector, then v = (v 1 , . . . , v n ).
Hence, the matrix complex conjugate is what we would expect: the same
matrix with all of its scalar components conjugated.
∗ hComplexConjugatei
created: h2013-03-21i by: hakrownei version: h31508i Privacy
setting: h1i hDefinitioni h12D99i h30-00i h32-00i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
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2
Properties of the Complex Conjugate
2.1
Scalar Properties
If u, v are complex numbers, then
1. uv = (u)(v)
2. u + v = u + v
−1
3. u
= u−1
4. (u) = u
5. If v 6= 0, then ( uv ) = u/v
6. Let u = a + bi. Then uu = uu = a2 + b2 ≥ 0 (the complex modulus).
7. If z is written in polar form as z = reiφ , then z = re−iφ .
2.2
Matrix and Vector Properties
Let A be a matrix with complex entries, and let v be a complex row/column
vector.
Then
T
1. AT = A
2. Av = Av, and vA = vA. (Here we assume that A and v are compatible
size.)
Now assume further that A is a complex square matrix, then
1. trace A = (trace A)
2. det A = (det A)
−1
3. A
= A−1
2