hermitian conjugate: meaning, definition, pronunciation and examples
C2Technical/Formal
Quick answer
What does “hermitian conjugate” mean?
The complex conjugate transpose of a matrix.
Audio
Pronunciation
Definition
Meaning and Definition
The complex conjugate transpose of a matrix.
In mathematics and quantum mechanics, an operator or matrix obtained by taking the transpose and the complex conjugate of each element of the original matrix. For an operator, it is the adjoint with respect to an inner product, especially in finite-dimensional spaces.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or grammatical differences. Both variants use the term identically in technical contexts.
Connotations
None beyond the strict technical meaning.
Frequency
Equally rare and specialised in both dialects, confined to advanced academic and professional STEM fields.
Grammar
How to Use “hermitian conjugate” in a Sentence
The hermitian conjugate of [matrix/operator]To find/take/calculate the hermitian conjugateVocabulary
Collocations
Examples
Examples of “hermitian conjugate” in a Sentence
verb
British English
- One must hermitian-conjugate the operator to find its adjoint.
- The procedure hermitian-conjugates the entire matrix.
American English
- You need to Hermitian-conjugate the matrix before proceeding.
- The algorithm automatically Hermitian-conjugates the input.
adverb
British English
- The matrices were transformed hermitian-conjugately.
- The operator acts hermitian-conjugately on the state.
American English
- The function was applied Hermitian-conjugately.
- The property holds Hermitian-conjugately for the pair.
adjective
British English
- The hermitian-conjugate properties are fundamental.
- We examined the hermitian-conjugate relationship.
American English
- The Hermitian-conjugate operator was computed.
- These are Hermitian-conjugate matrices.
Usage
Meaning in Context
Business
Never used.
Academic
Core terminology in linear algebra, functional analysis, quantum mechanics, and signal processing courses and papers.
Everyday
Never used.
Technical
Essential in mathematical physics, quantum computing, electrical engineering (for complex matrices), and advanced numerical analysis.
Vocabulary
Synonyms of “hermitian conjugate”
Strong
Neutral
Weak
Vocabulary
Antonyms of “hermitian conjugate”
Watch out
Common Mistakes When Using “hermitian conjugate”
- Confusing it with the simple transpose (without complex conjugation).
- Assuming it's the same as the inverse.
- Using the term for real matrices where it reduces to just the transpose, thus mis-specifying.
FAQ
Frequently Asked Questions
In finite-dimensional spaces with the standard inner product, yes, the terms are often used interchangeably. In infinite dimensions or with different inner products, 'adjoint' has a broader definition.
Common notations include Aᴴ (superscript H), A† (dagger), and sometimes A* (though the star can also denote simple conjugation).
For a matrix with only real entries, the Hermitian conjugate reduces to the simple transpose, as the complex conjugate of a real number is itself.
Observable quantities (like position, momentum) are represented by Hermitian operators, which are equal to their own Hermitian conjugate. This guarantees their eigenvalues (measurement results) are real numbers.
The complex conjugate transpose of a matrix.
Hermitian conjugate is usually technical/formal in register.
Hermitian conjugate: in British English it is pronounced /hɜːˈmɪʃ.ən ˈkɒn.dʒə.ɡət/, and in American English it is pronounced /hɝːˈmɪʃ.ən ˈkɑːn.dʒə.ɡət/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “None”
Learning
Memory Aids
Mnemonic
Think 'H' for Hermitian and 'H' for 'flip and conjugate' – you flip the matrix over its diagonal (transpose) and change all numbers to their complex mirror images (conjugate).
Conceptual Metaphor
A sophisticated mirroring operation: it reflects the matrix over its main diagonal and simultaneously reflects each number over the real axis on the complex plane.
Practice
Quiz
What is the Hermitian conjugate of a complex number a+bi?