Math 2270-004 Week 15 otes We will ot ecessarily iish the material rom a give day's otes o that day We may also add or subtract some material as the week progresses, but these otes represet a i-depth outlie o what we pla to cover These otes cover material or Moday I'll add course review material or Tuesday later Mo Apr 23 72 Secod derivative test, ad maybe aother coic diagoalizatio example Aoucemets: Warm-up Exercise:
From last week Spectral Theorem Let A be a symmetric matrix The all o the eigevalues o A are real, ad there exists a orthoormal eigebasis B u 1, u 2, u cosistig o eigevectors or A Eigespaces with dieret eigevalues are automatically orthogoal to each other I ay eigespace has dimesio greater tha 1, its orthoormal basis may be costructed via Gram Schmidt Diagoalizatio o quadratic orms: Let Q x i, j 1 or a symmetric matrix A, with real etries A symmetric orthoormal eigebasis B u 1, u 2, u a i j x T A x For the correspodig orthogoal matrix P u 1 u 2 u by the spectral theorem there exists a D P T A P, where D is the diagoal matrix o eigevalues correspodig to the eigevectors i P Ad we have x P y where y x B ad P P E B Thus Q x x T A x y T P T AP y y T D y i 1 i y 2 i So by the orthogoal chage o variables all cross terms have bee removed Applicatios iclude coic curves, quartic suraces, multivariable secod derivative test, pricipal compoet aalysis (PCA) i statistics, sigular value matrix decompositio (SVD) i geometry ad computer sciece, ad more
Deiitio: The quadratic orm Q x positive deiite i i, j 1 a i j x T A x (or A a symmetric matrix) is called Q or all We see that this is the same as sayig that all o the eigevalues o A are positive Deiitio: The quadratic orm Q x egative deiite i i, j 1 a i j x T A x (or A a symmetric matrix) is called Q or all We see that this is the same as sayig that all o the eigevalues o A are egative
First ad secod derivative tests rom multivariable calculus, revisited It turs out that a lot o multivariable calculus is easier to uderstad oce you kow liear algebra This is just oe example o where that happes (Math majors will see this, ad quite a bit more, i Math 3220) Let : x 1 x 1, x 2,, x x 2 : x, u a uit vector The d D u x dt 0 t u t 0 is the rate o chage o i the directio o u, at ("The directioal derivative o, at, i the directio o u" This geeralizes pure partial derivatives, which are rates o chage i the stadard coordiatedirectios) Usig the multivariable versio o the chai rule we compute d dt t u i 1 t u d d t, i t u i t u u So at t 0 this rate o chage is computed via D u u i 1 t u u i Deiitio: Let be a dieretiable uctio as above The is a critical poit or i ad oly i x 1, x 1, x 0 I other words, a critical poit is a poit at which all directioal derivatives are zero Local extrema o dieretiable uctios will oly occur at critical poits, but ot all critical poits are the locatios o local extrema The ways i which thigs ca go wrog are more iterestig tha i the sigle-variable case, where we used the secod derivative test
I your irst multivariable calculus class you were probably show a secod derivative test or uctios o (oly) two variables The oe you were probably show obscures what's really goig o, which is actually simpler to uderstad i geeral oce you kow liear algebra Here's what you were probably show (take rom the begiig o the Wikipedia article o this topic): https://ewikipediaorg/wiki/secod_partial_derivative_test
Cotiuig the discussio about directioal derivatives, Deiitio: Let,, u be as above The secod derivative o at, i the u directio is deied by D u u d 2 dt 2 t u t 0 We compute this expressio with the chai rule, startig with our expressio or the irst directioal derivative, rom the previous pages: d 2 dt 2 i 1 t u d dt i 1 i 1 j 1 d dt t u u i t u u i 2 t u u j u i At t 0 ad recallig that 2 D u u d 2 dt 2 2 this reads t u t 0 i 1 j 1 2 u i u j Deiitio: The Hessia matrix o at, D 2 is the (symmetric) matrix o secod partial derivatives; etry i j D 2 2 xi : x1 x 1 x1 x 2 x1 x D 2 x2 x 1 x2 x 2 x2 x : : x x 1 x x 2 x x So, D u u u T D 2 u
Theorem The uctio is cocave up i every directio u at i ad oly i the Hessia matrix D 2 is positive deiite The uctio is cocave dow i every directio u at i ad oly i the Hessia matrix D 2 is egative deiite The irst case happes i ad oly i all o the eigevalues o D 2 are positive, ad the secod case happes i ad oly i they are all egative I is a critical poit or, the i the irst case is a local miimum value; ad i the secod case it is a local maximum value I the Hessia has some egative ad some positive eigevalues, the is either a local miimum or a local maximum I all the eigevalues are o-egative, or i they are all o-positive, but some are zero, the urther work is required to determie whether is a local extreme value
Exercise 1 Explai the (more complicated) secod derivative test you were taught i multivariable calculus or uctios o just two variables, as a special case o the the more geeral oe that uses eigevalues Hit: has roots det xx xy yx yy xx yy xx yy 2 2 xx yy xx yy xy 2 2 4 xx yy xy 2
Exercise 2) Which o the ollowig uctios has a local miimum at the origi, i ay? Could you diagoalize the associated quadratic orms ad sketch level sets? 2a) x, y x 2 4 x y y 2 2b) x, y x 2 x y y 2
> with plots : plot3d x 2 4 x y y 2, x 1 1, y 1 1 ; plot3d x 2 x y y 2, x 1 1, y 1 1 ; >