4 syllables: ei, gen, val, ue. Stress on ei.
EYE-guhn-val-yoo
/ˈaɪɡəˌnvælju/
eigenvalue is pronounced EYE-guhn-val-yoo (/ˈaɪɡəˌnvælju/). It has four syllables (ei-gen-val-ue), with the stress on "ei". An eigenvalue is a scalar associated with a linear transformation that, when the transformation is applied to a vector, results in the vector being scaled but not rotated. In linear algebra, eigenvalues reveal intrinsic properties of a matrix, such as stability and inherent modes, and are found by solving the characteristic equation. They are fundamental to understanding diagonalizable systems and spectral decompositions.
An eigenvalue is a scalar associated with a linear transformation that, when the transformation is applied to a vector, results in the vector being scaled but not rotated. In linear algebra, eigenvalues reveal intrinsic properties of a matrix, such as stability and inherent modes, and are found by solving the characteristic equation. They are fundamental to understanding diagonalizable systems and spectral decompositions.
- You: You might first mispronounce the initial vowel as a long /eɪ/ instead of the correct /aɪ/. Correct by stressing the diphthong in EI: say /aɪ/ clearly, then glide into /ɡən/ without adding extra vowels. - You: The /ən/ in the middle can become a swallowed neutral vowel or disappear. Practice by saying EI-geh-n, articulating a light /ə/ then a crisp /n/. - You: The final /juː/ can be reduced to /uː/ or split as 'val-yoo' with weak coupling. Keep the final /juː/ strong and avoid turning it into a plain /uː/.
"The eigenvalue problem Ax = λx arises frequently in physics and engineering."
"For a symmetric matrix, eigenvalues are real and can be ordered from smallest to largest."
"The eigenvalues give the natural frequencies in a system’s mode analysis."
You say /ˈaɪ.ɡənˌvæl.juː/ (US/UK/AU share the same core pronunciation). Stress falls on the first syllable: EI-gen-value, with a light, quick /ən/ schwa in the middle and a clear /væl/ for the 'val' part. The final 'value' is stressed lightly on the first syllable of that word, sounding like 'VAL-yoo'. In connected speech, the /ən/ can be reduced, yielding /ˈaɪ.ɡə(n).ˈvæl.juː/. Audio reference: listen to mathematical vocab on Pronounce or Forvo for native speaker variants.
Common errors: (1) Misplacing the stress by over-emphasizing the second syllable; keep primary stress on EI. (2) Slurring /ɡən/ into a single sound; pronounce /ɡən/ as a quick but distinct ‘geh-n’ before /væl/. (3) Mispronouncing /væl/ as /val/ with a plain /æ/; aim for a short, tense /æ/ near /ɛ/ but not /æ/. Correct it by breaking the word into three beats: EI-gen-value, ensure the /ɪ/ in gen is like 'in' and the /væl/ is crisp but not overextended.
US/UK/AU share the EI primary stress, with /ˈaɪ/ as the onset. Subtle differences: US tends to reduce the /ən/ more and smooth the transition to /væl/; UK may have slightly lengthened /eɪ/ in EI and a more clipped /ɡən/; Australian often shows a lighter /ɡən/ and a more centered vowel in /ə/ depending on speaker. All three typically maintain /væl.juː/ at the end. IPA references align as /ˈaɪ.ɡənˌvæljuː/ with minor vowel shifts.
Two main challenges: the initial 'ei' digraph often carries a diphthong /ˈaɪ/ that can be misarticulated as /eɪ/; and the /ˈɡən/ cluster after /ˈaɪ/ can blur due to rapid speech. The final /væljuː/ combines a heavy /væl/ with /juː/, and English speakers sometimes separate into 'val-yoo' or compress it awkwardly. Focus on a crisp /ɡ/ release, a clear /ən/ before /væl/, and keep the final /juː/ distinct.
Yes. The stress pattern is fixed: EI-gen-value with primary stress on EI. Don’t shift stress to the end; keep the middle syllable light. The hallmark is the transition from /ˈaɪ/ to /ɡən/ and then a concise /ˈvæl.juː/; many learners fluff the /g/ or blend /n/ into the following /v/. Practice with slow syllable-timed articulation, then speed up while maintaining distinct syllables.
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- US: emphasize rhotics but keep /ɪ/ in gen short; slight reduction in /ən/ is common. - UK: maintain crisp /ɡən/ and slightly tighter /ə/ in the middle; listen for non-rhotic vowel nuances before /r/ sounds (not applicable here). - AU: flatter intonation, sometimes heavier /ə/ in the middle; ensure final /juː/ is not clipped. Always keep IPA references: /ˈaɪ.ɡənˌvæljuː/ and adapt to vowel shifts like /æ/ vs /ə/ depending on speaker.
The term eigenvalue comes from German eigen, meaning ‘own’ or ‘characteristic’, reflecting the idea that the scalar λ is an intrinsic characteristic of a linear transformation. The word eigen dates back to 19th-century linear algebraists Wilhelm Karl Johann Jordan and others who coined terms in German mathematics; the suffix -value aligns with mathematical values like eigenvector and invariant value. The concept emerged as matrices were studied more abstractly, with early work by Camille Jordan and Carl Runge influencing its usage. In English, the term was popularized in the late 19th and early 20th centuries as matrix theory matured, and eigenvalue became the standard descriptor for the scaling factors associated with eigenvectors. First known English usage appears in mathematical treatises that translate or adapt German terminology into accessible notation for a widening audience in physics and engineering. Over time, the concept has become central in stability analysis, quantum mechanics, vibration theory, and data science, where eigenvalues often reveal core system properties and natural modes of behavior.
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Words that rhyme with "eigenvalue"
-eve sounds
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