What does consimilarity mean?
Consimilarity is a specialised term drawn from linear algebra, where it names a precise relationship between two square matrices: one matrix is consimilar to another if it can be transformed into that matrix's complex conjugate by an invertible change of basis. In other words, it extends the familiar notion of matrix similarity by folding in conjugation, which makes it useful when working with complex-valued systems, canonical forms, and problems involving real representations of complex structures. The word is built transparently from Latin roots — con- ('together') plus similis ('like') — so its form signals its meaning: a being alike together, or mutual resemblance. Beyond mathematics, consimilarity occasionally appears in formal prose to describe close reciprocal likeness between ideas or objects, though such use is rare. Readers will most often meet it in research papers and advanced textbooks, where its exact technical definition matters and casual paraphrase risks error.
nounA relation between two square matrices in which one can be obtained from the other by multiplication on both sides by a matrix and the conjugate of its inverse — equivalently, similarity of one matrix to the conjugate of the other. More broadly, resemblance between things; a being jointly or mutually similar.
- A technical relation in linear algebra: two square matrices A and B are consimilar if A = S B S̄⁻¹ for some invertible matrix S, i.e. A is similar to the conjugate of B.
- Rare general use: mutual or shared resemblance between things.
"The two Hermitian matrices exhibit consimilarity, so they share many spectral properties."
"Every real square matrix is consimilar to its own transpose, a fact exploited in canonical-form proofs."
"The essay traced a consimilarity between the two philosophical traditions despite their different origins."
The plural is rare and typically appears in mathematical writing discussing multiple instances of the relation.
"The paper catalogued the consimilarities among the candidate canonical forms."
In linear algebra, two matrices can be 'similar' — but only consimilar matrices are alike after you've also flipped them into their mirror-image complex conjugates.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of consimilarity
Consimilarity derives from Latin similis, meaning 'like' or 'similar', combined with the prefix con-, meaning 'with' or 'together'. The same root produced English 'similar', 'similarity', 'assimilate', and 'simulate', making these the word's closest cognates. The term arose as a technical coinage within mathematics, formed by analogy with 'consimilar' and modelled on the established term 'similarity', as matrix theorists needed a name for the conjugate analogue of ordinary similarity.
Related word forms
How consimilarity is actually used
Almost exclusively found in advanced mathematics, particularly matrix theory and complex linear algebra, where it appears in formal writing rather than speech. Outside that context it is vanishingly rare and likely to be unfamiliar to general readers; the looser sense of mutual resemblance is uncommon and largely confined to older or literary prose.
Easily confused with consimilarity
Similarity of matrices uses only an invertible transform, while consimilarity additionally involves complex conjugation; as ordinary nouns, 'consimilarity' stresses mutual or shared resemblance and is far rarer.
Consanguinity means descent from a common ancestor (blood relation), not resemblance.