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Lecture6.1: Whatwewillnot betalkingabout Optimization and Computational Linear Algebra for Data Science Lo Miolane Warning / Home works 1 exams . 1/6 The determinant There exists a function det : R n n R called the determinant

  1. Lecture6.1: Whatwewillnot betalkingabout Optimization and Computational Linear Algebra for Data Science Léo Miolane

  2. Warning / Home works 1 exams . 1/6

  3. ⇒ The determinant There exists a function det : R n × n → R called the determinant that = = verifies det( M ) = 0 is not invertible. M ⇐ ⇒ The determinant can be computed using the following formula: over all som n " - ÿ Ÿ det( M ) = ‘ ( ‡ ) M i, σ ( i ) " rrorqdeu :L - than - Maru , Mano i =1 σ ∈ S n y - - , - - numbers 1,2 - - n ← TG ) - g - 1 n=4 Exe : 2341 is , a oak Yatra ) depending ah t . modeling of . - - 4 t . a 2/6

  4. ⇐ ⇒ Geometrical interpretation £ 2 /detlA=ad-bI A=f ¥? dD ) Vz I detest A = - bet = lad o - CA ) , o del off A - o - lin dep Va , vz ⇐ s . . not invertible A is 3/6

  5. ⇒ ⇒ ⇐ Link with eigenvalues eigenvalue of A X rot Kala - did ) is an that Ao exists there such vto . - AID ) f { o } Kala invertible - XII not A is ⇐ s . det ( A - kid ) O = - function of that we write X 4/6

  6. The characteristic polynomial polynomial • Pala ) is in se a . A = ( ta z ) Let's consider Ed : det ( FIFA ) - aid ) det CA Pala ) = = - 2 =lnZ-3at1T ( n - a) ( 2- a) = characteristic polynomial of A called the Pa is • eigenvalues of A its roots the are . deg CPA ) f n most n distinct has at hence A i • eigenvalues 5/6 .

  7. Example take A=fy ; - Ro - Tsing I 1 - ÷ : ) ; Let's - for D= Tk ' i Pala ) det CA - aid ) i . = = Ia2tI det ( IIe ) = For all ) = htt 71 70 Pala a ER • , any real eigenvalues not Hence A- does have . PACE ) a. Ztt C- 1) t 1 = 0 = • = oiheeisothae ) ( complex eigenvalue of A - i is a . 6/6

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