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Fundamental operation on complex numbers
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but with opposite
Complex_conjugate
Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle
Conjugate_transpose
Operation in complex geometry
In complex geometry, the complex conjugate line of a straight line is the line that it becomes by taking the complex conjugate of each point on this line
Complex_conjugate_line
Theorem about polynomials
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P
Complex conjugate root theorem
Complex_conjugate_root_theorem
Mathematics concept
In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} that
Complex conjugate of a vector space
Complex_conjugate_of_a_vector_space
representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate vector space V as follows:
Complex conjugate representation
Complex_conjugate_representation
Number with a real and an imaginary part
real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying
Complex_number
Roots of an algebraic element's minimal polynomial
generalizes complex conjugation, since the algebraic conjugates over R {\displaystyle \mathbb {R} } of a complex number are the number itself and its complex conjugate
Conjugate element (field theory)
Conjugate_element_(field_theory)
Conjugate homogeneous additive map
: V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if f ( x + y ) = f ( x ) + f ( y ) (additivity)
Antilinear_map
Matrix equal to its conjugate-transpose
self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose. That is, if the element in the j
Hermitian_matrix
"Bouncing back" of waves at an interface
the aberrating optics a second time. If one were to look into a complex conjugating mirror, it would be black because only the photons which left the
Reflection_(physics)
Complex-differentiable (mathematical) function
{\displaystyle f} with respect to z ¯ {\displaystyle {\bar {z}}} , the complex conjugate of z {\displaystyle z} , is zero: ∂ f ∂ z ¯ = 0 , {\displaystyle
Holomorphic_function
Mathematical operation in linear algebra
denotes the conjugate transpose of x {\displaystyle \mathbf {x} } (conjugate of the transpose, or equivalently transpose of the conjugate). Matrix multiplication
Matrix_multiplication
Critical point where a periodic solution arises
system of differential equations. When this matrix has a pair of complex-conjugate eigenvalues that cross the imaginary axis as a parameter is varied
Hopf_bifurcation
other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation
Complex_representation
Vector space with generalized dot product
of F. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted 0 {\displaystyle \mathbf
Inner_product_space
Distance from zero to a number
{\displaystyle |z|=r.} Since the product of any complex number z {\displaystyle z} and its complex conjugate z ¯ = x − i y {\displaystyle {\bar {z}}=x-iy}
Absolute_value
Polynomial equation of degree 3
non-real complex conjugate roots. This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is
Cubic_equation
Topics referred to by the same term
identifies equivalent dynamical systems Conjugate words in combinatorics Harmonic conjugate in complex analysis Convex conjugate, the ("dual") lower-semicontinuous
Conjugation
Function of the coefficients of a polynomial that gives information on its roots
has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if
Discriminant
Reals with an extra square root of +1 adjoined
1} . A split-complex number has two real number components x and y, and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ =
Split-complex_number
Algebraic operation on coordinate vectors
{b_{i}}}},} where b i ¯ {\displaystyle {\overline {b_{i}}}} is the complex conjugate of b i {\displaystyle b_{i}} . When vectors are represented by column
Dot_product
Number with an integer power equal to 1
exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also an nth root of unity: 1 z = z − 1 = z n − 1 = z ¯ . {\displaystyle
Root_of_unity
Complex exponential in terms of sine and cosine
when used in multiplication or powers of complex numbers. Any complex number z = x + iy, and its complex conjugate, z = x − iy, can be written as z = x +
Euler's_formula
Horizontal line used in mathematical notation
1428571428571428571... a + b i ¯ {\displaystyle {\overline {a+bi}}} complex conjugate Y = A B ¯ {\displaystyle Y={\overline {AB}}} boolean NOT (A AND B)
Vinculum_(symbol)
Type of vector space in math
{x^{2}+y^{2}}}\,.} The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w: ⟨ z , w ⟩ = z w ¯ . {\displaystyle
Hilbert_space
Theorem in electrical engineering
maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance. The mathematics of the theorem also applies
Maximum power transfer theorem
Maximum_power_transfer_theorem
Polynomial equation of degree two
double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included
Quadratic_equation
Adjusting input/output impedances of an electrical circuit for some purpose
Z_{\text{load}}=Z_{\text{source}}^{*},} where a superscript * indicates the complex conjugate. A conjugate match is different from a reflection-less match when either
Impedance_matching
Number whose cube is a given number
real, one of the cube roots is real and the two other are nonreal complex conjugate numbers. Otherwise, the three cube roots are all nonreal. For example
Cube_root
Property determining comparison and ordering
Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, z ¯ {\displaystyle {\bar
Magnitude_(mathematics)
Type of complex function
mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable
Hermitian_function
2nd-degree plane curve which is reducible
single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines). All these degenerate
Degenerate_conic
the Hermitian conjugate (also called the conjugate transpose) of A {\displaystyle A} , defined as applying both the complex conjugate and the transpose
+_h.c.
Conjugate transpose of an operator in infinite dimensions
ft(A^{-1}\right)^{*}} Conjugate linearity: (A + B)∗ = A∗ + B∗ (λA)∗ = λA∗, where λ denotes the complex conjugate of the complex number λ "Anti-distributivity":
Hermitian_adjoint
Notation for quantum states
versa. The Hermitian conjugate of a complex number is its complex conjugate. The Hermitian conjugate of the Hermitian conjugate of anything (linear operators
Bra–ket_notation
Property of a differential manifold that includes complex structures
with its complex conjugate is the zero section: L ∩ L ¯ = 0 {\displaystyle L\cap {\overline {L}}=0} ; L is maximal isotropic, i.e. its complex rank equals
Generalized_complex_structure
Polynomial function of degree 4
follows: If ∆ < 0 then the equation has two distinct real roots and two complex conjugate non-real roots. If ∆ > 0 then either the equation's four roots are
Quartic_function
mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle
Complex_Lie_algebra
Concepts from linear algebra
The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs. The spectrum of a matrix is the
Eigenvalues_and_eigenvectors
Sum in algebraic number theory
character χ the equation relating L(s, χ) and L(1 − s, χ) (where χ is the complex conjugate of χ) involves a factor[clarification needed] G ( χ ) | G ( χ ) |
Gauss_sum
Inequality between integrals in Lp spaces
measurable real- or complex-valued functions defined on S. If ‖fg‖1 is finite, then the pointwise products of f with g and its complex conjugate function are
Hölder's_inequality
Polynomial with reversed root positions
complex numbers, when p ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n , {\displaystyle p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n},} the conjugate reciprocal
Reciprocal_polynomial
Measure of the electric polarizability of a dielectric material
'-i\varepsilon ''} is the complex permittivity Note that this is using the electrical engineering convention of the complex conjugate ambiguity; the physics/chemistry
Permittivity
Relativistic wave description of fermions
relates a four-component spinor to its charge conjugate. As a 2×2 differential equation acting on a complex two-component spinor, resembling the Weyl equation
Majorana_equation
Non-tensorial representation of the spin group
+ i v {\displaystyle z=u+iv} is complex, and z ¯ = u − i v {\displaystyle {\bar {z}}=u-iv} is the complex conjugate of z {\displaystyle z} . Then H 2
Spinor
Curve from a cone intersecting a plane
conic section are real, the points at infinity are either real or complex conjugate. What should be considered as a degenerate case of a conic depends
Conic_section
Dual pair of vector spaces
{\displaystyle H.} Let H ¯ {\displaystyle {\overline {H}}} denote the complex conjugate vector space of H , {\displaystyle H,} where H ¯ {\displaystyle {\overline
Dual_system
Characteristic property of holomorphic functions
suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of z {\displaystyle z} , denoted z ¯ {\textstyle {\bar {z}}} , is
Cauchy–Riemann_equations
Linear transformation of spacetime coordinates
be a 2 × 2 complex matrix with determinant 1 and let A † {\displaystyle A^{\dagger }} be the hermitian conjugate of A (the complex conjugate of the transpose
Biquaternion Lorentz transformation
Biquaternion_Lorentz_transformation
{\overline {\Box }}} 1. Complex conjugate: If z is a complex number, then z ¯ {\displaystyle {\overline {z}}} is its complex conjugate. For example, a + b
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Notation in general relativity
chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward
Newman–Penrose_formalism
Measure of the opposition to current flow by an external electrical load
canceling out the reactance. When this occurs the circuit is said to be complex conjugate matched to the signals impedance. Note this only maximizes the power
Input_impedance
derived from the two different conventions. The two definitions are complex conjugates of each other. One way to incorporate attenuation into the mathematical
Mathematical descriptions of opacity
Mathematical_descriptions_of_opacity
Covariance and correlation
dt} where f ( t ) ¯ {\displaystyle {\overline {f(t)}}} denotes the complex conjugate of f ( t ) {\displaystyle f(t)} , and τ {\displaystyle \tau } is called
Cross-correlation
Statement in number theory
Finally, if each zero ρ {\displaystyle \rho } comes paired with its complex conjugate ρ ¯ {\displaystyle {\bar {\rho }}} , then we may combine terms to
Li's_criterion
Physical characteristic of oscillating systems
pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general, Complex conjugate pairs of poles
Resonance
Complex matrix whose conjugate transpose equals its inverse
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U
Unitary_matrix
Relativistic wave equation describing massless fermions
transformation. The skew complex conjugate ω ψ ∗ = i σ 2 ψ {\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi } can be recognized as the charge conjugate form of ψ
Weyl_equation
Type of representation in representation theory
act either on real or complex column vectors. A real representation on a complex vector space is isomorphic to its complex conjugate representation, but
Real_representation
Polynomial equation of degree 4
real solutions – then there is another complex solution x {\displaystyle x} 2 which is the complex conjugate of x {\displaystyle x} 1. If the other two
Quartic_equation
Vector operation
=\mathbf {u} \mathbf {v} ^{\dagger }.} The conjugate transpose of a matrix results from taking the complex conjugate of each entry in the transpose of a matrix
Outer_product
Matrix operation which flips a matrix over its diagonal
^{\text{T}}=-\mathbf {A} .} A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline)
Transpose
Quadric surface with one axis of symmetry and no center of symmetry
hyperbolic if the factors are real; elliptic if the factors are complex conjugate. An elliptic paraboloid is shaped like an oval cup and has a maximum or
Paraboloid
Probability distribution on complex matrices
column p-vector of random complex Gaussian zero-mean samples and ( . ) H {\displaystyle (.)^{H}} is an Hermitian (complex conjugate) transpose. If the covariance
Complex_Wishart_distribution
Irreducible representation of the rotation group SO
fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical
Wigner_D-matrix
Movement of an object which leaves at least one point unchanged
{\displaystyle {\bar {v}}^{\text{T}}{\bar {v}}} is real, it equals its complex conjugate v T v {\displaystyle v^{\text{T}}v} , and v ¯ T v {\displaystyle {\bar
Rotation
Mathematical description of fermions
two component spinors transform differently under the two distinct complex-conjugate spin-1/2 representations of the Lorentz group. This pairing is of
Dirac_spinor
Mathematical function, denoted exp(x) or e^x
{\displaystyle e^{i\pi }=-1} and the functional identity. The complex conjugate of the complex exponential is e z ¯ = e z ¯ . {\displaystyle {\overline
Exponential_function
Values which describe behavior of a linear electric circuit
is the complex conjugate of Z i {\displaystyle Z_{i}} , V i {\displaystyle V_{i}} and I i {\displaystyle I_{i}} are respectively the complex amplitudes
Scattering_parameters
Change of the sign of a square root
2026-01-17. "Conjugate in Math - Surds, Complex Number, Rationalization". Cuemath. Retrieved 2026-01-17. "Conjugate in Math - Surds, Complex Number, Rationalization"
Conjugate_(square_roots)
Differential form on a manifold which is permitted to have complex coefficients
differentials of the holomorphic coordinates with q differentials of their complex conjugates. The ensemble of (p, q)-forms becomes the primitive object of study
Complex_differential_form
Horizontal line immediately above a portion of writing
-2+0.07918={\bar {2}}.07918} The overline notation can indicate a complex conjugate and analogous operations. if x = a + i b {\displaystyle x=a+ib} ,
Overline
Generalization of complex inner products
{\overline {a}}} is the complex conjugate of a scalar a . {\displaystyle a.} A complex sesquilinear form can also be viewed as a complex bilinear map V ¯ ×
Sesquilinear_form
Mathematical relation defining a sequence
instead there may be some that are complex numbers. In the latter case, all the complex roots come in complex conjugate pairs. If all the characteristic
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Here, λ ¯ {\displaystyle {\overline {\lambda }}} denotes the complex conjugate of the complex number λ {\displaystyle \lambda } , and similarly for K ( x
Fredholm's_theorem
Filters used in signal processing that are optimal in some sense
unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal
Matched_filter
Formula that provides the solutions to a quadratic equation
the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other. Geometrically, the roots represent the
Quadratic_formula
Array of numbers
conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex
Matrix_(mathematics)
Bijective antilinear map between two complex Hilbert spaces
1 {\displaystyle H_{1}} , where the horizontal bar represents the complex conjugate. If additionally one has H 1 = H 2 {\displaystyle H_{1}=H_{2}} then
Antiunitary_operator
Group representation
the complex conjugate of the transpose, the transpose is the conjugate of the adjoint. Thus, ρ ∗ ( g ) {\displaystyle \rho ^{\ast }(g)} is the complex conjugate
Dual_representation
Number which when multiplied by x equals 1
z ¯ ‖ z ‖ {\displaystyle {\tfrac {\bar {z}}{\|z\|}}} gives us the complex conjugate with a magnitude reduced to a value of 1, so dividing again by |z|
Multiplicative_inverse
form, a specific sesquilinear form Hermitian function, a complex function whose complex conjugate is equal to the original function with the variable changed
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Every polynomial has a real or complex root
product of a polynomial and its complex conjugate (obtained by replacing each coefficient with its complex conjugate). A root of this product is either
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Topics referred to by the same term
a complexity class in computational complexity theory Complex conjugate, an operation on complex numbers, commonly abbreviated as c. c. Cc (space group)
CC
Mathematical function that preserves angles
U {\displaystyle U} . If f {\displaystyle f} is antiholomorphic (complex conjugate to a holomorphic function), it preserves angles but reverses their
Conformal_map
vanishes. If E is a complex vector bundle, then the conjugate bundle E ¯ {\displaystyle {\overline {E}}} of E is obtained by having complex numbers acting
Complex_vector_bundle
Number, approximately 1.618
{\bigr )}} , because the sum of any fifth root of unity and its complex conjugate, z + z ¯ {\displaystyle z+{\bar {z}}} , is a golden integer, a
Golden_ratio
Triangle in hyperbolic geometry
denotes the complex conjugate of α {\displaystyle \alpha } and, similarly, β ¯ {\displaystyle {\overline {\beta }}} denotes the complex conjugate of β {\displaystyle
Hyperbolic_triangle
Method of drawing geometric objects
and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π)
Straightedge and compass construction
Straightedge_and_compass_construction
Length in a vector space
denotes its conjugate transpose. This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner
Norm_(mathematics)
Mathematical operator in quantum optics
displacement in optical phase space, α ∗ {\displaystyle \alpha ^{*}} is the complex conjugate of that displacement, and a ^ {\displaystyle {\hat {a}}} and a ^ †
Displacement_operator
Mandelbar Set
defined similarly to the Tricorn fractal, but instead of squaring the complex conjugate of z, we raise it to any positive integer degree, d: The tricorn T
Tricorn_(mathematics)
Equation from stability analysis
identity matrix and A ¯ {\displaystyle {\bar {A}}} is the element-wise complex conjugate of A {\displaystyle A} . One may then solve for vec ( X ) {\displaystyle
Lyapunov_equation
Typographical symbol (*)
a superscript The complex conjugate of a complex number (the more common notation is z ¯ {\displaystyle {\bar {z}}} ). The conjugate in a composition algebra
Asterisk
Measure used in radio engineering and telecommunications
Impedance matching is achieved when the source impedance is the complex conjugate of the load impedance. The easiest way of achieving this, and the
Standing_wave_ratio
referred to different bases. A matrix is consimilar to itself, its complex conjugate, its transpose and its adjoint matrix. Every matrix is consimilar
Matrix_consimilarity
Mathematics concept
applications of these ideas. Almost complex manifold Complex manifold Complex differential form Complex conjugate vector space Hermitian structure Real
Linear_complex_structure
Function of propagation delay and Doppler frequency
}s(t)s^{*}(t-\tau )e^{i2\pi ft}\,dt} where ∗ {\displaystyle ^{*}} denotes the complex conjugate and i {\displaystyle i} is the imaginary unit. Note that for zero
Ambiguity_function
Concept in mathematics
{\displaystyle f(z)} of the complex variable z := x + i y ∈ Ω . {\displaystyle z:=x+iy\in \Omega .} That is, v {\displaystyle v} is conjugate to u {\displaystyle
Harmonic_conjugate
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