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Hints

8. AA−−→′ =3−−→AM/2. 14. cos2 θ 1. ≤ 19. Rotate the regular n-gon through 2π/n. 22. Take X = A first. 23. −−→AB = −−→OB −→OA. − 24. If a + b + c + d = 0, then 2(b + c)(c + d)= a2 b2 + c2 d2. − − 25. XA2 = −−→OX −→OA 2 =2R2 2 −−→OX, −→OA if O is the center. | − | − h i 26. Consider projections of the vertices to any line. 27. Project faces to an arbitrary plane and compute signed areas. 35. Compute (cos θ + i sin θ)3. 37. Divide P by z z with a remainder. − 0 40. (z z )(z z¯ ) is real. − 0 − 0 41. Apply Vieta’s formula. iθ 45. e± = cos θ i sin θ. ± 46. See Figure 13. n 50. Show first that k equals the number of k-element subsets in the set of n objects.  52. Show that the sum of distances to the foci of an ellipse from points on a tangent line is minimal at the tangency point; then reflect the source of light about the line.

54. e = P ′P / P ′Q is the slope of the secting plane. | | | | 55. Apply the forthcoming classification of quadratic curves. 56. AX2 + CY 2 is even in X and Y . 57. Q( x, y)= Q(x, y) only if b = 0. − 60. cos θ =4/5. 62. Rotating the coordinate system as in Example on p. 11, don’t forget to apply the change to the linear terms.

201 202 Hints 63. Q( x, y)= Q(x, y) for a quadratic form Q. − − 65. Reflect the plane about a suitable line. 68. x2 +4xy = (x +2y)2 4y2. − 69. Examine when the quadratic equation at2 +2bt + c = 0 has a double root. 76. Given that A and B can be transformed into C, find a way to transform A into B. 80. For each of the normal forms, draw the region of the plane where the quadratic form is positive. 81. Draw the region on the plane where xy > 0. 2 2 85. First apply the Inertia Theorem to make Q = x1 + x2. 91. Take x1 = x, x2 =x ˙. 97. In the “toy version,” one of the quadratic forms is x2 +y2, the squared Euclidean distance to the origin. 98. 4ac = (a + c)2 (a c)2. − − 2 102. 3n(n+1) n2 = n(n+3) > n . 2 − 2 2 103. 1 = 0. 6 105. If ba = 1 and ab′ = 1, then b = bab′ = b′. 110. This is a rephrasing of the previous exercise. 114. If (a,n) = 1, then ka + ln = 1 for some integers k and l. 116. Show that a non-zero homomorphism between two fields maps 1 to 1, and inverses to inverses. 118. Construct the homomorphism f : Z K such that f(1) = 1. → 119. Take F = 0, 1,η,ζ , and set 1+1=0, η + ζ = 1, ηζ = 1. { } 120. x2 + x +1=(x + η)(x + ζ).

121. Compute 0 + 0′. 122. u = ( 1)u. − − 123. Redefine multiplication by scalars as λu = 0 for all λ K and all ∈ u . ∈V 124. 0u = (0 + 0)u =0u +0u. 129. Rotated parallelograms are still parallelograms. 132. A linear function on is uniquely determined by its restrictions V⊕W to and . 135.V TranslateW given linear subspaces by any vector in the intersection of the affine ones. 141. Examine the kernel of the map R[x] R2 which associates to a → polynomial P the pair (P (1), P ( 1)) of its values at x = 1. − ± 152. cos 4θ + i sin 4θ = (cos θ + i sin θ)4. Hints 203 154. Prove that E is injective by identifying with Kn and evaluating V coordinate functions on a non-zero vector. f 158. To a linear form f : / K, associate π / K. V W → V →V W → 160. F = pn, where n is the dimension of F as a Z -vector space. | | p 167. Both associate to an input x X the output w = f(g(h(x)). ∈ 176. Pick two 2 2-matrices at random. × 179. Compute BAC, where B and C are two inverses of A. 180. Given two isomorphisms B : and A : , find an U →V V →W isomorphism from to . W U 181. Consider 1 2 and 2 1 matrices. 1 × × 183. Solve D− AC = I for D and C. 187. Which 2 2-matrices are anti-symmetric? × 191. Start with any non-square . 201. Each elementary product contains a zero factor. 204. There are 12 even and 12 odd permutations. 1 2 ... n − 1 n 206. Permutation has length n .  n n − 1 . . . 2 1  2 1 208. Pairs of indices inverted by σ and σ− are the same.  209. The transformation: logarithm lagorithm algorithm consists → → of two transpositions. 212. Locate a pair (i,i + 1) of nearby indices which σ inverses, compose σ with the transposition of these indices, and show that the length decreases by 1. 217. 247 = 2 100+4 10 + 7. · · 221. det( A) = ( 1)n det A. − − 223. det(CtQC) = (det Q)(det C)2. 234. Apply the 1st of the “cool formulas.” 235. Compute the of a 4 4-matrix the last two of whose × rows repeat the first two. 236. Apply Binet–Cauchy’s formula to the 2 3 matrix whose rows are × xt and yt.

237. Show that ∆n = an∆n 1 + ∆n 2. − − m 239. Use the defining property of Pascal’s triangle: k = m 1 m 1 k− + k −1 . −  240. Use the fact of algebra that a polynomial in (x1,...,xn), which vanishes when x = x , is divisible by x x . i j i − j 241. Divide the kth column by k and apply Vandermonde’s identity. 243. H : R7 R1. → 204 Hints 244. The space spanned by columns of A and B contains columns of A+B. t n m 252. (A a)(x)= a(Ax) for all x K and a (K )∗. ∈ ∈ 266. Modify the Gaussian elimination algorithm of Section 2 by permuting unknowns (instead of equations). 267. Apply the LUP decomposition to M t. 1 268. Apply LPU, LUP , and PLU decompositions to M − . 273. When K has q elements, each cell of dimension l has ql elements. 277. Consider the map Rn Rp+q defined by the linear forms. → 290. T = A + iB where both A and B are Hermitian.

291. x, y = z†w. h i 293. z, w = z†w. h i 294. Use the previous exercise. 296. The normal forms are: i Z 2 i Z 2, Z 2 Z 2, Z 2. | 1| − | 2| | 1| − | 2| | 1| 301. Take the linear form for one of new coordinates. 303. Prove that 1 is a non-square. − 306. There are 1 even and 3 odd non-degenerate forms. 310. 1 =1 2 . 3 − 3 311. Start with inverting 2-adic units 1. ( is a wild card). ···∗∗∗ ∗ 313. Use the previous exercise. 337. Apply Vieta’s formula. 339. Compute x, Ax for x = t v , where v is an orthonormal basis h i i i { i} of eigenvectors, and t2 = 1. i P 340. If AB = BA, thenP eigenspaces of A are B-invariant. 345. Find an eigenvector, and show that its orthogonal complement is in- variant. 349. Construct v1,..., vn by applying the Spectral Theorem to the non- negative Hermitian operator A†A, and check that Av corresponding i ∈W to distinct nonzero eigenvalues µi of A†A are pairwiseperpendicular. 350. Rephrase the previous exercise in matrix form. 353. The equation x + ty, x + ty = 0 quadratic in t has no real solutions h i unless x is proportional to y. 355. x, x = ( x )2 + x2. h i i i t 357. det(U U)P = 1. P 366. See Example 1. 376. Use the previous exercise. 378. Prove first that the radius of the circle must be equal to the middle semiaxis of the ellipsoid. Hints 205 381. For “exactly onne” claim, use the coordinate-less description of the spectrum provided by the minimax principle. 382. When x 0, sin x x. ≈ ≈ n 385. If T denotes the cyclic shift of coordinates in C , then C = C0I + n 1 C1T + + Cn 1T − . ··· − 398. Use Newton’s binomial formula. 411. Pick A and B so that eAeB = eBeA. 6 414. Verify it first for a Jordan cell. 419. Make two of the lines the coordinate axes on the plane, and the third one the graph of an isomorphism between them. 420. If the first three lines are y = 0, x = 0, and y = x., then the fourth one is y = λx. 421. If d = 0, a = , b = 1, then λ = c. ∞ 423. Assuming that the spaces have the same dimension, and the maps Ui between them are invertible, consider the determinant (or eigenvalues) of the composition of the maps around the cycle. 426. Use the theory of Bruhat cells. 427. Use Lemma, and apply induction on the length of the shortest of the legs. 428. Use Sylvester’s rule together with the previous exercise. 431. On the unit sphere of area 4π, the area of a triangle with the angles ( π , π , π ) has the area π + π + π π. Use this to compute the number of p q r p q r − required triangles.

433. Compute Rv1 Rv2 to show that it is a Jordan cell of size 2 with the eigenvalue 1. 434. They correspond to positive roots e e = v + + v , where i − j i+1 ··· j 0 i < j n. ≤ ≤

Answers

1. mg/2, mg√3/2. 2. 18 min (reloading time excluded). 6. It rotates with the same angular velocity along a circle centered at the barycenter of the triangle formed by the centers of the given circles. 10. 3/4.

13. 7−→OA = OA−−→′ +2OB−−→′ +4OC−−→′ for any O. 15. 3/2. 17. 2 u, v = u + v 2 u 2 v 2. h i | | − | | − | | 18. (b) No. 28. Yes; Yes (0); Yes. 1+5i 1 i√3 29. (a) 13 ; (b) −2 . 2 31. z − . | | 33. (a) √2, π/4; (b) 2, π/3. − − 1+i√3 34. − 2 . 1 √5 √6 i√2 38. 2 i; i ± ; 1+ i, 1+ i; 1 − . ± 2 ± 2 √3 i √3 i √3 i √3 i 39. 2, 1 i√3; i, ± − ; i√2, i√2; ± , ± ; i, ± . − ± 2 ± ± 2 2 ± ± 2 k 42. ak = ( 1) 1 i1< 0. 49. cos 3θ =4 cos3 θ 3 cos θ, sin 3θ =3 sin θ 4 sin3 θ. − − 58. Yes, k(x2 + y2). 59. y = x; level curves of (a) are ellipses, of (c) hyperbolas. ± 60. 2X2 Y 2 = 1. − 62. Semiaxes 2/√3 and 2.

207 208 Answers 64. (a) ( α2 β2, 0), (b) ( α2 + β2, 0). ± − ± 68. 2nd ellipse,p 1st & 4th hyperbolas.p 72. A pair of intersecting lines, x 1= (2y 1). − ± − 79. For normal forms, X2 and 0 can be taken. 84. (n + 1)(n + 2)/2. 88. x(t)= x(0)eλt. 3t t 89. x(t)= x(0)e ,y(t)= y(0)e− ,z(t)= z(0). 92. X˙ = iX , X˙ = iX . 1 1 2 − 2 96. Yes, some non-equivalent equations represent the empty set. n(n+1) 101. 2 . 107. Use the Euclidean algorithm: dividea by b with the remainder r, then divide b by r with the remainder r1, then divide r by r1 with the remainder r2, etc. until the remainder becomes 0. 108. Use the previous exercises. 111. n. 113. 1, 3, 5, 7. 134. Points, lines, and planes. 137. pn; pmn. n(n 1)/2 138. p − . 140. (b) Ker D = K[xp] K[x]. ⊂ 149. dim = 2. 151. xk = xkL (x)+ + xk L (x), k =0,...,n 1. 1 1 ··· n n − 155. (a) n; (b) n(n + 1)/2. 156. 2, if n> 1. 161. yes, no, no, yes, no, yes, yes, no.

163. E = va, i.e. eij = viaj , i =1,...,m, j =1,...,n. cos θ sin θ 164. . sin θ −cos θ   166. Check your answers using associativity, e.g. (CB)A = C(BA).

3 1 0 2 Other answers: BAC = , ACB = . 3 3 3− 6    

cos 1◦ sin 1◦ 172. − . sin 1◦ cos 1◦   k 173. If B = A , then bi,i+k = 1, and all other bij = 0. 174. The answer is the same as in the previous problem. Answers 209 175. (a) If A is m n, B must be n m. (b) Both must be n n. × × × 177. Exactly when AB = BA. 184. Restriction of linear functions on to the subspace . W V 186. I. 188. S =2x y + x y + x y , A = x y x y . 1 1 1 2 2 1 1 2 − 2 1 189. No, only if AB = BA. 192. No; yes (of x, y R1); yes (in R2). ∈ 193. ( xi)( yi) and i=j xiyj /2. 6 195. PT † = Pz2 w1; S1 P= (z1 w2 + z2 w1)/2, S2 = (z1 w2 z2 w1)/2i; − H = (z z + z z )/2, H = (z z z z )/2i. 1 1 2 2 1 2 1 2 − 2 1 196. (z1 w2 + + zn 1 wn + z2 w1 + + zn zn 1)/2. ··· − ··· − 197. T C†TD. 7→ 199. No, yes, n2. 202. 1; 1; cos(x + y). 203. 2, 1; 2. − 205. k(k 1)/2. − 207. n(n 1)/2 l. − − 211. Transpositions τ (i) of nearby indices (i,i + 1), i =1,...,n 1. − 213. E.g. τ14τ34τ25. 214. No; yes. 215. + (6 inverted pairs); + (8 inverted pairs). 216. 1 522 200; 29 400 000. − − 218. 0. 219. Changes sign, if n =4k +2 or n = 4k +3 for some k, and remains unchanged otherwise. 220. x = a1,...,an. 222. Leave it unchanged. n(n 1)/2 226. (a) ( 1) − a a a , (b) (ad bc)(eh fg). − 1 2 ··· n − − 227. (a) 9, (b) 5. 5 7 1 18 18 18 1 1 0 − 1 5 7 − 228. (a) 18 18 18 , (b) 0 1 1 .  7 − 1 5  " 0 0− 1 # 18 18 − 18 229. (a) x = [ 2 , 1 , 1 ]t, (b) x = [1, 1, 1]t. − 9 3 9 − n 1 231. (det A) − , where n is the size of A. 232. Those with 1. ± 233. (a) x1 =3, x2 = x3 = 1, (b) x1 =3, x2 =4, x3 = 5. 234. (a) x2 x2 , (b) (ad bc)3. − 1 −···− n − 210 Answers

n n 1 238. λ + a1λ − + + an 1λ + an. ··· − 239. 1. 241. 1!3!5! (2n 1)!. ··· − 243. 1.

245. The system is consistent whenever b1 + b2 + b3 = 0. 251. (a) Yes, (b) yes, (c) yes, (d) no, (e) yes, (f) no, (g) no, (h) yes. 255. 0 codim k. ≤ ≤ 257. Two subspaces are equivalent if and only if they have the same di- mension. 258. The equivalence class of an ordered pair , of subspaces is deter- U V mined by k := dim U, l := dim , and r := dim( + ), where k,l r n V U V ≤ ≤ can be arbitrary. n n n n 2 n n 1 259. (a) q ; (b) (q 1)(q q)(q q ) (q q − ); n n n − 2 −n r −1 ··· − (c) (q 1)(q q)(q q ) (q q − ); n − n − −n ···r 1 −r r r r 1 (d) (q 1)(q q) (q q − )/(q 1)(q q) (q q − ). − − ··· − − − ··· − 260. x =3, x =1, x = 1; inconsistent; x =1, x =2, x = 2; 1 2 3 1 2 3 − x =2t t , x = t , x = t , x = 1; 1 1 − 2 2 1 3 2 4 x = 8, x = 3+ t, x = 6+2t, x = t; x = x = x = x = 0; 1 − 2 3 4 1 2 3 4 x = 3 t 13 t , x = 19 t 20 t , x = t , x = t ; 1 17 1 − 17 2 2 17 1 − 17 2 3 1 4 2 x = 16+ t + t +5t , x = 23 2t 2t 6t , x = t , x = t , x = t . 1 − 1 2 3 2 − 1 − 2 − 3 3 1 4 2 5 3 261. λ = 5. 262. (a) rk=2, (b) rk=2, (c) rk=3, (d) rk=4, (e) rk=2. 263. Inverse matrices are: 1 1 1 1 2 1 0 0 1 4 3 − − − 1 1 1 1 1 3 200 1 5 3 , − − , − . − − 4  1 1 1 1   31 19 3 4  " 1 6 4 # − − − − − 1 1 1 1 23 14 2 3  − −   − −      n 271. 2 l(σ), where l(σ) is the length of the permutation. − n q 1 272. [n]q! := [1]q[2]q [n]q (called q-factorial), where [k]q := q −1 . ··· − 274. Inertia indices (p, q)=(1, 1), (2, 0), (1, 2). 279. Empty for p = 0, has 2 components for p =1, and 1 for p =2, 3, 4. 2 2 2 2 2 2 2 283. z1 + z2 =1,z1 + z2 =0,z2 = z1 ,z1 =1,z1 = 0. 284. Two parallel lines. 289. Those all of whose entries are imaginary. 290. T (z, w)= 1 [T (z + w, z + w)+ iT (iz + w,iz + w) (1 + i) (T (z, z)+ T (w, w))]. 2 − 297. dii = ∆i/∆i 1. − 298. (p, q)=(2, 2), (3, 1). Answers 211 316. No. 317. The sum of the squares of all sides of a parallelogram is equal to the sum of the squares of the diagonals. 330. id. 354. θ(ei, ei+1)=2π/3, while all other pairs are perpendicular.

356. One can take fi = ei e0, and hi = ei ei 1, where i =1,...n. − − − 358. Rotations about the origin, and reflections about a line passing thor- ough the origin, which have determinants 1 and 1 respectively. − 364. 1. ± 0 1 371. The operator is . 1 0  −  10 0 2 0 0 3 0 0 397. (a) 02 0 (b) −0 1 0 (c) −0 1 1 " 0 0 1 # " 0 0 1 # " 0 0 1 # 2 1 0 − 2 00 1 0 0 10 0 (e) 0 2 1 (f) 0 1 0 (k) −0 0 1 (l) 0 i 0 . " 0 0 2 # " 0− 00 # " 0 0 0 # " 0 0 i # − 407. (a n(a a /3))3n. 0 − 0 − 1 408. (a) (1 + √10)/3. 0 1 0 0 411. A = and B = will do. 0 0 1 0     418. Pairs of m n-matrices; pairs of matrices of transposed sizes. × 422. K1. 424. The restriction to the eigenline. 429. (a) (2, 2, r), (2, 3, 3), (2, 3, 4), (2, 3, 5); (b) (3, 3, 3), (2, 4, 4), (2, 3, 6). 431. Symmetry planes of regular polyhedron (tetrahedron in case (b), oc- tahedron or cube in case (c), and icosahedron or dodecahedron in case (d)) partition the surface of the sphere circumscribed around the polyhedron into (respectively, 24, 48, and 120) required spherical triangles. 432. Since v, v = 2, we have: h i x x, v v, y y, v v = x, y x, v v, y y, v x, v +2 x, v y, v h −h i −h i i h i−h ih i−h ih i h ih i = x, y . h i 434. i,j : 0 K1 ≃ ≃ K1 0 ... (i zeroes followed by V ··· → → → · · · → → → j i copies of K1, followed by n j zeroes). − − 0,n 1,n n 1,n 435. − . V ⊕V ⊕···⊕V tA tA 436. X(t)= e X(0)e− . 437. Eigenvalues are λ λ , i, j =1,...,n. In the space of linear forms, i − j they form the root system of type An 1. Eigenvectors are matrices Eij − whose all entries are 0 except a single 1 in the ith row and jth column.

Index

n-space, 43 block , 79 n-vectors, 43 Bruhat cell, 113 absolute value, 12 canonical form, 29 addition of vectors, 39 canonical isomorphism, 56 additivity, 7 canonical projection, 49 adjoint form, 70 Cartesian coordinates, 9 adjoint map, 65, 96 category of vector spaces, 44 adjoint systems, 99 Cauchy, 89 adjugate matrix, 84 Cauchy – Schwarz inequality, 9, 135 affine subspace, 45, 49 Cauchy’s interlacing theorem, 156 algebraic number, 40 Cayley transform, 139 algebraically closed field, 170 characteristic, 42 annihilator, 57 characteristic equation, 140 anti-Hermitian form, 71, 122 characteristic polynomial, 35, 163 anti-Hermitian quadratic form, 122 Chebyshev, 55 anti-linear, 70 Chebyshev polynomials, 55 anti-symmetric , 68 classification theorem, 29 anti-, 68 codimension, 98 anti-symmetric operator, 151 cofactor, 83 argument of complex number, 13 cofactor expansion, 83, 87 associative, 62 column, 47 associativity, 4 column space, 105 , 102 commutative square, 95, 189 axiom, 39 commutativity, 4 axis of symmetry, 25 commutator, 124 complementary multi-index, 87 B´ezout, 17 complete flag, 111 B´ezout’s theorem, 17 completing squares, 26 back substitution, 102 complex conjugate, 11 barycenter, 4 complex conjugation, 149 basis, 5, 51 complex multiplication, 153 bijective, 44 complex sphere, 120 bilinear form, 67 complex vector space, 40 bilinearity, 8 complexification, 149 Binet, 89 components, 47 Binet–Cauchy formula, 89 composition, 61 block, 79 congruent modulo n, 41

213 214 Index conic section, 21 elementary row operations, 101 conics, 119 ellipse, 22 connected graph, 187 ellipsoid, 157 continuous time, 175 entry, 47 coordinate Euclidean space, 147 equivalent, 29 coordinate flag, 112 equivalent conics, 119 coordinate system, 5 equivalent linear maps, 95 coordinate vectors, 43, 47 equivalent representations, 188 coordinates, 5, 43, 47, 53 Euclidean algorithm, 208 Cramer’s rule, 85 Euclidean inner product, 147 cross product, 90 Euclidean space, 147 cross-ratio, 193 Euclidean structure, 147 cylinder, 119 Euler’s formula, 18 evaluation, 60 Dandelin, 22 evaluation map, 48 Dandelin’s spheres, 21 even form, 129 Darboux, 130 even permutation, 75 Darboux basis, 130 exponential function, 17 decomposable representation, 190 degenerate bilinear form, 155 Fibonacci, 175 Descartes, 9 Fibonacci sequence, 175 determinant, 37, 73 field, 12, 40 , 47 field of p-adic numbers, 132 , 169 finite dimensional spaces, 52 dimension, 52 flag, 111 dimension of Bruhat cell, 113 focus of ellipse, 22 direct sum, 32, 44, 190 focus of hyperbola, 23 direct sum of representations, 190 focus of parabola, 23 directed segment, 3 Fourier, 160 directrix, 23 Fourier basis, 160 discrete dynamical systems, 175 Fundamental Formula of Mathemat- discrete time, 175 ics, 19 discriminant, 14, 133 distance, 135 Gabriel, 191 distributive law, 12, 60 Gaussian elimination, 101 distributivity, 4 Gelfand’s problem, 155 dot product, 7 golden ratio, 176 dot-product, 67 , 137 dual basis, 54 Gram–Schmidt process, 126, 137 dual map, 65 graph, 45, 187 dual space, 46 gravitation constant, 162 greatest common divisor, 41 eccentricity, 23 group, 198 edge, 187 eigenspace, 141, 163 half-linear, 70 eigenvalue, 33, 140, 163 Hamilton–Cayley equation, 171 eigenvector, 140, 163 harmonic oscillator, 159 elementary product, 73 Hasse, 132 Index 215 head, 3 Jordan cell, 168 Hermite, 70 , 168 Hermite polynomials, 55 Jordan system, 35 Hermitian adjoint, 138 Hermitian adjoint form, 70 kernel, 44 Hermitian conjugate matrix, 70 kernel of form, 118, 121 Hermitian form, 71 kinetic energy, 158 Hermitian inner product, 135 Hermitian isomorphic, 136 Lagrange, 87 Hermitian operator, 138 Lagrange polynomials, 55 Hermitian quadratic form, 71, 121 Lagrange’s formula, 87 Hermitian space., 135 law of cosines, 9 Hermitian-anti-symmetric form, 70 LDU decomposition, 110 Hermitian-symmetric form, 70 leading coefficient, 102 Hilbert space, 135 leading entry, 102 homogeneity, 7 leading minors, 124 homogeneous system, 98 left inverse, 96 homomorphism, 41 length, 135 homomorphism theorem, 50 length of permutation, 75 hyperbola, 22 linear combination, 4 hyperplane, 100 linear dynamical systems, 175 hypersurface, 119 linear form, 45, 59 linear function, 45, 59 , 63 linear map, 44 identity permutation, 75 linear ODE, 175 imaginary part, 11 linear recursion relation, 175 imaginary unit, 11 linear subspace, 43 inconsistent system, 103 linear transformation, 64 indecomposable representation, 190 linearly dependent, 52 indices in inversion, 75 linearly independent, 52 induction hypothesis, 53 Linnaeus, 29 inertia index, 117 lower triangular, 108 Inertia Theorem, 28 lower-triangular matrix, 47 injective, 44 LPU decomposition, 108 inner product, 7 LU decomposition, 110 invariant subspace, 141 LUP decomposition, 111 inverse matrix, 63 inverse transformation, 64 M¨obius band, 56 inversion of indices, 75 mathematical induction, 53 involution, 149 matrix, 47, 59 irreducible representation, 191 matrix entry, 59 isometric Hermitian spaces, 136 matrix product, 60, 61 isomorphic spaces, 44 metric space, 135 isomorphism, 44 Minkowski, 132 iterations of linear maps, 175 Minkowski–Hasse theorem, 132 , 83, 87 Jordan block, 35 multi-index, 87 Jordan canonical form, 168 multiplication by scalar, 3 216 Index multiplication by scalars, 39 potential energy, 158 multiplicative, 12 power of matrix, 65 multiplicity, 14 principal axes, 25, 157 principal minor, 117 nilpotent operator, 165 projection, 136 non-degenerate bilinear form, 155 Pythagorean theorem, 9 non-degenerate Hermitian form, 124 nontrivial linear combination, 52 q-factorial, 114, 210 normal form, 29 quadratic curve, 21 normal operator, 138, 151 quadratic form, 23, 31, 69 null space, 44, 105 quadratic formula, 14 quiver, 187 odd form, 129 quotient space, 49 odd permutation, 75 ODE, 175 range, 44 operator, 138 rank, 31, 93 opposite coordinate flag, 112 rank of linear system, 98 opposite vector, 39 rank of matrix, 94 oriented graph, 187 real normal operators, 148 orthogonal, 136 real part, 11 orthogonal basis, 115 real spectral theorem, 151 orthogonal complement, 141 real vector space, 40 orthogonal diagonalization, 145 realification, 149 Orthogonal Diagonalization Theorem, reduced , 102 155 reducible representation, 191 orthogonal projection, 137 reflection, 198 orthogonal projector, 139 regular nilpotent, 165 orthogonal transformation, 147 representation, 187 orthogonal vectors, 9 right inverse, 96 orthonormal basis, 125, 136, 147 root of unity, 15 root space, 164 parabola, 27 root system, 199 partition, 167 row echelon form, 102 Pascal’s triangle, 203 row echelon form of rank r, 102 pendulum, 162 row space, 105 permutation, 73 , 108 scalar, 39, 40 phase plane, 159 scalar product, 7 pivot, 102 semiaxes, 157 Pl¨ucker, 90 semiaxis of ellipse, 26 Pl¨ucker coordinates, 98 sesquilinear form, 70, 121 Pl¨ucker identity, 90 sign of permutation, 73 PLU decomposition, 111 similarity, 163 polar, 13 similarity transformation, 64 polar decomposition, 146 simple problems, 37 positive definite, 116 simple quiver, 191 positive operator, 145 singular value decomposition, 145 positivity, 7 span, 51 Index 217 spectral theorem, 140 vector subspace, 43 spectrum, 33, 155 vector sum, 3 , 47, 63 vertex, 187 square root, 15, 145 Vieta, 17 standard basis, 51 Vieta’s theorem, 16 standard coordinate flag, 111 standard coordinate space, 43 Young tableaux, 167 standard Euclidean space, 71 standard Hermitian space, 71 zero representation, 190 subrepresentation, 191 zero vector, 3, 39 subspace, 43 surjective, 44 Sylvester, 124 symmetric bilinear form, 68, 115 symmetric matrix, 68 symmetric operator, 151 symmetricity, 7 system of linear equations, 61 tail, 3 tautology, 50 time-independent dynamical system, 175 total anti-symmetry, 77 trace, 139 transition matrix, 63 transposed form, 68 transposed map, 65 transposed matrix, 66 transposed partition, 167 transposition matrix, 109 transposition permutation, 75 triangle inequality, 9, 135 , 168 unipotent matrix, 110 unit coordinate vectors, 51 unit vector, 8 unitary rotation, 143 unitary space, 135 unitary transformation, 139 upper triangular, 108 upper-triangular matrix, 47

Vandermonde, 91 Vandermonde’s identity, 91 vector, 3, 39 vector space, 36, 39