פירוק פולרי

במתמטיקה, פירוק פולרי (או פירוק UH או פירוק UP) של מטריצה היא הכללה מסוימת של הצגת מספר מרוכב בהצגה פולרית: $z=re^{i\theta }\,$ .

הגדרה ומשפט הפירוק

כל מטריצה ריבועית A מרוכבת ניתן לכתוב בצורה:

A=UP

, (או

A=UH

)

כאשר U היא מטריצה אוניטרית, ו-P (או H) היא מטריצה הרמיטית חיובית. כאשר המטריצה A הפיכה, הפירוק יחיד, ו-P חיובית לחלוטין.

קשר לשם "פירוק פולרי" ניתן לראות בעובדה ש:

\det P=r=|\det A|

\det U=e^{i\theta }

כאשר $r$ ו- $\theta$ ממשיים.

ולכן:

\det A=\det U\,\det P=re^{i\theta }

קשר לפירוק לערכים סינגולריים

קיים קשר הדוק בין פירוק זה לפירוק לערכים סינגולריים.

A={\tilde {U}}\Sigma V^{*}=\underbrace {{\tilde {U}}V^{*}} _{U}\underbrace {V\Sigma V^{*}} _{P}=UP

לקריאה נוספת

Higham, Nicholas J. (1986). "Computing the polar decomposition with applications". SIAM J. Sci. Stat. Comput. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics. 7 (4): 1160–1174. doi:10.1137/0907079. ISSN 0196-5204.

קישורים חיצוניים

פירוק פולרי, באתר MathWorld (באנגלית)

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