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Unformatted text preview: Prof. Dancis MATH 401 Exam 3 April 30, 2004 Heed these instructions: Use efﬁcient methods for calculations. Calculations should be labelled and easily
understood. Laptops and hand calculators are not permitted. Each time that the theorem about solutions of (all) homogeneous linear equations is
used (implicitly or explicitly) in your calculations during this test, indicate this by stating
“theorem about homo. lin eq., at the point that the theorem is used. Same for theorem
about solutiOns of (all) nonhomogeneous linear equations Use this info to shortcut calculations. Credit will be deducted for not using this
info. If M:(“ b c d (‘5 51) (i)=<a+b>(i) (Z 3) (31)=<ab><31) 1.(a) State the spectral theorem about real symmetric or hermitian matrices. 4 points ), then A1” = detIM (:10 :lb), whenever .det .M ;A 0. (c) Prove that every square stochastic matrix has one as an eigenvalue. State (without
proof) the two other propositions that you use. 8 points (d) Prove that the product of two 7 x 7 unitary matrices is a unitary matrix. 6 points (e) Given that matrix, lbs diagonalizable and given that matrix, N is _s_i_1_r_1ilar to
matrix, I”; prove that matrix, N is also diagonalizable and has the same eigenvalues as
matrix, (ll/I. State (without proof) the other propositions/theorems that you use. 7 points (f) Prove that the complex eigendata for real matrices occurs in complex conjugate
pairs. 6' points (g) Let .M, P and D be constant 7 x 7 matrices; Al and P are invertible. Let
v = v(t) E R7 x R, be a vector valued function of time. Let W : w(t) = Pv(t) and Ill—1K = P_1DP(1,71.(11\I\"/(t)+ Kv(t) = 0. Show that + Dw(t) = 0. 10 points
(h) Given that AI is a 3 x3—matrix, with top row (0. 3, 4). Suppose that v : (:r. y. z)T,
with 1‘ 2 y 2 z > 0, is an eigerwector of AI associated with eigenvalue A. Show that,
A g 7. 8 points OVER 2. Find a series solution, for u(:r,t), on the domain 0 g :r 3
system of equations: i, and t 2 0, to this a.” = 9m and 11.1(0, t) = 0 = 114%. t) (Reminder: uI = 2—2) Do NOT skip steps in your calculations. 1.9 points
State the eigenvalue problem in a box. 2 points
Is zero an eigenvalue? Why or why not? You may assume that there are NO negative
eigenvalues. 3 points 3. DO NOT SOLVE this system of equations:
u” = 7m + 8sin 2x and u(0, t) = 7 and u(7r, t) = 9 and u(m, 0) = —2 sin 22 + 7 + 7sin 9x + 5sin12x
on the domain 0 g :r 3 71', and t 2 0.
(a) Write this system in linear equation form. 2 points
(b) Write down the system of associated homogeneous equations. 3 points (c) Given that two functions, u(:c,t) and v(:r,t) satisfy the equations:
um 2: 7m + 8sin 2:1: and 11$; = 711; + 8sin2cc. Show that w(:r, t) = u(:c, t) — v(x, t) satisﬁes the associated homogeneous equation. 2
points ((1) Suppose that. some of the given data for u(:z,t) was obtained by measurements
or tolerances. Call u($,f.), the calculated solution. Let v(.r.,t) be the real solution.
Speciﬁcally, suppose that 11" 1 2
u(0, t) — v(0,t) g — and t) — v(7r,t) S —, V90
10 10 ‘
3
and lu(:r,0) — v(a;,0) S m, VOSIS7T
Find a good bound on t) — 'u($, t)], VQO and US$377. State any theorem or corollary
that you use. 10 points
4. Given vn : fly/112,14, Vnez+. Show/prove that 114 = 111400. 4 points
5. Graph y — 5x = i(5y — x)2 in the my—plane. State the change—of —eoordinate matrix.
7 points
How many hours per week do you study for this course? 1 points Total 100 points [\3 C005, 0 's cum*1: 3.—
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V3; Mvz Prof. Dancis l\/IATH 401 Exam 3 Some Answers April 30, 2004 1.(a) State the spectral theorem about real symmetric or hermitian matrices. S : O_1DO, where D is a real diagonal matrix and O is a real orthogonal matrix. QR S = O‘lDO : OTDO, where D is a real diagonal matrix. Crucial that you state that D is a real (diagonal) matrix, and either state or indicate
that O is a real orthogonal matrix. Or Theorem V1129 (c) Prove that every square stochastic matrix has one as an eigenvalue. State (without
proof) the two other propositions that you use. This is Proposition V1.29. (d) Prove that the product of two 7 X 7 unitary matrices is a unitary matrix.
Given: A and B are unitary matrices, that is A‘1 : 14* and B—1 = To prove: AB is a unitary matrix, that is (143)“1 2 (AB)*.
Proof. (.AB)‘1 : 3414—1 : B*A* : (AB)*. \/ " ,1 ,‘ ‘L (e) Given that matrix, 114 is diagonalizable and given that matrix, N is similar to
matrix, M'; prove that matrix, N is also diagonalizable and has the same eigenvalues as
matrix, MI State (without proof) the other propositions/theorems that you use. This is Proposition V1.3.14. (f) Prove that the complex eigendata for real matrices occurs in complex conjugate
pairs. This is Proposition V1.73. (g) Let 111, P and D be constant 7 X 7 matrices; JW and P are invertible. Let
v : v(t) E R7 X R, be a vector valued function of time. Let w : w(t) : Pv(t) andll/[_1K = P“1DP and MW) + Kv(t) : 0.
Show that + Dw(t) : 0. 10 pu‘L'rLtS Pay? Mm) + Kv(t) = 0 => 0 : wt) +A/[_1Kv(t) = {/(t) + PAlDPv.
Multiply by P on the left yields: 0 : PV(t) + DPv : we) + Dw(t) y. (h) Given that [W is a 3 ><3—matrix, with top row (0, 3,4). Suppose that v : y, ﬂy],
with at 2 y 2 z > 0, is an eigenvector of NI associated with eigenvalue A. Show that A37. This is similar to Exercise V1.25, which was done in class as board work.
The eigenvalue/ vector equation is 11/1120 2 A120. Look at the top coordinate of this
equation: 3y+4z=mﬁxz33+4ig3+4=7 ¢
.71; I 2. Find a series solution, for u(1‘,t), on the domain 0 S 1' S i, and t 2 0, to this
system of equations:
u“. = 9m and u$(0, t) : 0 =u1.(ﬁ,t) (Reminder: um = Do NOT skip steps in your calculations. State the eigenvalue problem in a box. 1s zero an eigenvalue? Why or why not? You may assume that there are NO negative
eigenvalues. This is similar to Example 11.7.2. Some selected lines, which are modiﬁcations of
Example 11.7.2 follow: since the only way for g(:r) : h(t) is for both functions to be constant. The equations for X are the eigenvalue problem:
(a?) X”(.r) + /\X(1:) = 0 is a Basic Spring Block Equation. Its general solution is: L(X(:c)) : —X”(:r) : /\X with boundcﬁj conditions : X’(G) : O z ‘ : A cos ﬁzr + Bsin \/:\—CL', VA and B
X/(IL‘) : —A\/:\—sin Vim + Bﬁcos \/X..T7 VA and 3
Plug in the boundary condition:
0 = X’(0) : —A\/Xsin0 + B\/Xcos0 2 BA :9 B = o.
This simpliﬁes X’(:L') to:
X’(.7;) : vAﬁsin Plug in the boundary condition: 1 1 l .
0:X’(1): —A\/XsinZ\/X:> /\:0 or sinlﬁzo. 1 1 _
sing”: O :9 EVA = MT :3» /\n :— 4R7T7Vnez+ The eigenfunctions are X.n(a;) = cos4mr1',\‘/neg+.
To ﬁnd Tn(t): Tm) : —%AnTn(t) : —(4n7r)2Tn(t). q 2 —(~inr\21 Tn(t) = e 9 This is an Exponential Decay Equation. ——(4nt)2! un(.r,t) :2 Xn(:L')Tn(t) = e 9 cos 471:1, Vn=1,2,3,.... All linear combinations of solutions, to a system of homogeneous linear equations, are
more solutions. um t) = 2:11 um, t) = 2:1 ‘
Can ,\ = 0? The equation, X”(a;) + /\X(r£) = 0 will simplify to X”(:L') = 0 =>
X(a:) = C3: + D,VC and D. This, together with the bondary conditions will result in a
non—zero eigenvector Xo(:E) : 1. So yes, here, /\ = 0, with To(t) = (30‘ : 1. Hence
U0($,t) = X0($)To(t) = 1
Taking the linear combinations of no and u, above, yields: COS 4717193: Vconstants an, n=l,2,3, é° —(4'n7.)2L ane 9 COS 4n'7rxy Vconstants an, n=0,1,'2,3, 3. DO NOT SOLVE this system of equations:
um : Tut + 88in 2:1? and MD, : 7 and u(7r) t) : 9
and u(:z:, 0) : —2sin 21' + 7 + 7sin91r, + 5 sin 12:7; on the domain 0 S :1: g 7?, and t 2 O. (a) Write this system in linear equation form.
Linear—equations have the form7 L(v) : we, where wo
“input” variable v. is not a function of the L1(u) : um. — 7m, : 8 sin 21‘, Lg(u) : u(0, t) = '7,
L3(u) : u(7r, t) : 9
L4(U) : u(a;, 0) = #2 sin 21* + ‘7 + Tsin .92: + 5 sin 121:. (b) Write down the system of associated homogeneous equations. L100) : rum. 7 71m : O,
Lg(w) : 10(07 t) : 07
L3(’U}) : 10(711, t) : O.
L4(w) : 10(23, 0) : 0. (0) Given that two functions. u(:L', t) and v(;1:,t) satisfy the equations:
uxI : Tut + 8 sin 2:6 and UM : 7m + 88l1l217.
Show that t) : u(:L', t) — 11(2), t) satisﬁes the associated homogeneous equation. in“. ~ 7w: : [11(10) 2 L1(u) — L1(v) : 85in21‘ —— 8sin21L' 2: 0 That is7 the difference between two solutions of a non—homogeneous linear equation is a
solution to the associated homogeneous equation. (d) Suppose that some of the given data for was obtained by nmasurements
or tolerances. Call u(:t,t), the calculated solution. Let v(1‘,t) be the real solution. Speciﬁcally, suppose that l 2
u(0,t) —'U(O.,t) S E and fa(7r,’t) — U(7r,t)l 3 E, V/tio
duo) (on/3v
. LL‘ —/ "\ —— \ v./r:»r
an u , u , _ 10, O#w_ Find a good bound on
that you use. Similar to Example 11.8.7.
Answer. Set w(x,t) : u(:,t,t) — 'U(:c,t). Then w(:L',t) satisﬁes these equations and inequalities: Lﬁw) : wag,” — 7w; : l 2
w(0,t)l S E and l’iu(7t,t)l _<_’ 15, V20
dl < 0)! f 3 v
', w r' \ — 1. 7
an L, _ 10, 03 g The top equation is a heat equation, without a heat source. Hence the Maximal
Principle and its corollary apply to 10(1‘, t), The corollary say that when w(:L', t) satisﬁes
a heat equation, wit/tout a heat source, (10” : 6211.135) then upper and lower bounds of
w(.r,t), on the Sides and “bottom” of the domain are also upper and lower bounds of
w($, t) on the entire semi—inﬁnte strip of a domain. Here ‘13—0 ’1’. w(;t,t) ii %, for all points (at) on the sides and “bottom’7 of the
domain. Hence the corollary implies that —130 w(_:c,t for all points (:c,t) on
the entire semi—inﬁnte strip of the domain. The Maximal Principle and its corollary do NOT apply to a heat equation, with a heat source, like um. I Tut + 8sin 2.95. \ / ,,
l 4. Given an : 11/11)”; 1, Vnezir. Show/ prove that '04 = 114/4120. This is Proposition v1.4.1 5. Graph y — 5:1; : ::(5y — fl])2 in the xy—plane. State the change—of coordinate matrix.
1   1‘1 —1 5 ’1' _‘
Answer. Set yl : y — 01' and :51 = by — :L‘. Then : , . bo the
yr —0 l y
. . . —l " _ r change—of —coordmate matrix 18 <_5 y # 5:5 : i<oy — ZL'JZ becomes yl : :17, whose graph is tangent to the fol—axis at the origin.__ The $1—axis is the line 3} = 5:5; and
The yl—axis is the line 53/ : x. When the graph is moved to the zy—plane, it remains
tangent to the ccl—axis at the origin. ...
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 Spring '04
 Dancis
 Math, Linear Algebra, Algebra, Matrices, Orthogonal matrix, Normal matrix, Unitary matrix

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