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g(the)f(pap)q(er)h(is)f(as)h(follo)o(ws.)24 b(Section)17 b(2)h(in)o(tro)q(duces)f(the)g(de\014nition)g(of)g(the)h(SVD,)-46 1821 y(follo)o(w)o(ed)d(b)o(y)g(a)h(discussion)h(of)f(the)f(prop)q (erties)h(of)g(the)g(comp)q(onen)o(ts)f(of)h(the)g(SVD.)f(Section)g(3)i (explores)-46 1882 y(further)d(prop)q(erties)g(of)g(the)g(SVD)g(and)h (pro)o(vides)f(a)g(geometric)e(in)o(terpretation)i(of)g(the)g(singular) g(v)m(alues.)-46 1942 y(Section)i(4)i(lists)f(a)g(n)o(um)o(b)q(er)e(of) j(in)o(teresting)d(applications)j(and)f(Section)g(5)g(concludes)g(the)f (pap)q(er)i(with)-46 2002 y(a)f(discussion)f(of)h(the)f(adv)m(an)o (tages)h(and)g(disadv)m(an)o(tages)g(of)g(using)g(the)f(SVD.)-46 2168 y Fu(2)83 b(De\014nition)27 b(of)h(the)f(SVD)-46 2278 y Fr(In)12 b(this)h(section,)f(w)o(e)g(assume)g(a)h(familiarit)o (y)c(with)k(the)f(basic)h(terminology)e(of)i(linear)e(algebra,)j(and)f (refer)-46 2338 y(the)18 b(reader)g(to)g([And86])g(for)g(a)g(more)f (complete)f(co)o(v)o(erage.)25 b(W)l(e)18 b(restrict)f(our)i(atten)o (tion)e(to)i(matrices)-46 2398 y(of)g(real)f(n)o(um)o(b)q(ers)e(and)j (refer)f(the)g(reader)g(to)h([DD88)q(])f(for)g(a)h(discussion)f(of)h (the)f(SVD)g(using)h(complex)-46 2458 y(n)o(um)o(b)q(ers.)h(This)c (presen)o(tation)g(is)g(largely)g(adapted)h(from)e([FMM77)o(].)27 2519 y(Using)f(the)f(sup)q(erscript)h Fq(T)20 b Fr(to)15 b(denote)e(the)h(transp)q(ose)h(of)f(a)g(v)o(ector)f(or)i(matrix,)d(w)o (e)h(sa)o(y)h(t)o(w)o(o)g(v)o(ectors)-46 2579 y Fq(x)k Fr(and)g Fq(y)i Fr(are)d Fp(orthogonal)i Fr(if)f Fq(x)577 2561 y Fo(T)604 2579 y Fq(y)g Fr(=)f(0)p Fq(:)g Fr(In)h(t)o(w)o(o)g(or) g(three)f(dimensional)g(space,)g(this)h(simply)e(means)-46 2639 y(that)22 b(the)f(v)o(ectors)g(are)g(p)q(erp)q(endicular.)37 b(Let)21 b Fq(A)g Fr(b)q(e)h(a)f(square)h(matrix)e(suc)o(h)h(that)h (its)f(columns)f(are)917 2790 y(1)p eop %%Page: 2 2 2 1 bop -46 -45 a Fr(m)o(utually)16 b(orthogonal)k(v)o(ectors)d(of)i (length)f(1,)h(i.e.)26 b Fq(x)959 -63 y Fo(T)986 -45 y Fq(x)17 b Fr(=)g(1.)28 b(Then)19 b Fq(A)e Fr(is)i(an)f Fp(orthogonal)k(matrix)-46 16 y Fr(and)16 b Fq(A)85 -2 y Fo(T)112 16 y Fq(A)d Fr(=)h Fq(I)t Fr(,)h(the)g(iden)o(tit)o(y)e (matrix.)19 b(T)l(o)d(simplify)c(the)j(notation,)h(assume)f(that)h(a)f (matrix)f Fq(A)h Fr(has)h(at)-46 76 y(least)g(as)h(man)o(y)e(ro)o(ws)h (as)h(columns)e(\()p Fq(M)k Fn(\025)14 b Fq(N)5 b Fr(\).)27 136 y(A)19 b Fp(singular)24 b(v)m(alue)e(decomp)r(osition)d Fr(of)h(an)f Fq(M)g Fn(\002)13 b Fq(N)24 b Fr(matrix)18 b Fq(A)g Fr(is)h(an)o(y)h(factorization)f(of)g(the)-46 196 y(form)801 256 y Fq(A)13 b Fr(=)h Fq(U)5 b Fr(\006)p Fq(V)1016 236 y Fo(T)1043 256 y Fq(;)-46 340 y Fr(where)18 b Fq(U)24 b Fr(is)18 b(an)g Fq(M)g Fn(\002)12 b Fq(M)24 b Fr(orthogonal)c(matrix,)d Fq(V)29 b Fr(is)18 b(an)h Fq(N)f Fn(\002)12 b Fq(N)24 b Fr(orthogonal)19 b(matrix,)e(and)i(\006)f (is)g(an)-46 400 y Fq(M)c Fn(\002)8 b Fq(N)21 b Fr(diagonal)16 b(matrix)d(with)i Fq(s)606 407 y Fo(ij)650 400 y Fr(=)f(0)h(if)g Fq(i)e Fn(6)p Fr(=)h Fq(j)k Fr(and)e Fq(s)1022 407 y Fo(ii)1062 400 y Fr(=)d Fq(s)1136 407 y Fo(i)1164 400 y Fn(\025)h Fr(0.)21 b(F)l(urthermore,)13 b(it)i(can)g(b)q(e)g(sho)o (wn)-46 460 y(that)h(there)g(exist)f(\(non-unique\))g(matrices)f Fq(U)22 b Fr(and)16 b Fq(V)27 b Fr(suc)o(h)16 b(that)g Fq(s)1226 467 y Fm(1)1260 460 y Fn(\025)d Fq(s)1335 467 y Fm(2)1369 460 y Fn(\025)g(\001)8 b(\001)g(\001)14 b(\025)g Fq(s)1569 467 y Fo(N)1616 460 y Fn(\025)g Fr(0)i([GVL83)q(].)-46 520 y(Henceforth)i(w)o(e)h(will)f(assume)g(the)h(SVD)g(has)h(suc)o(h)f (a)h(prop)q(ert)o(y)l(.)29 b(The)20 b(quan)o(tities)e Fq(s)1578 527 y Fo(i)1611 520 y Fr(are)h(called)f(the)-46 581 y Fp(singular)k(v)m(alues)c Fr(of)g Fq(A)p Fr(,)f(and)i(the)e (columns)g(of)h Fq(U)23 b Fr(and)18 b Fq(V)29 b Fr(are)18 b(called)f(the)h(left)f(and)h(righ)o(t)f Fp(singular)-46 641 y(v)n(ectors)p Fr(,)e(resp)q(ectiv)o(ely)l(.)27 741 y(F)l(or)i(example,)c(the)j(matrix)f Fq(A)e Fr(=)664 668 y Fl( )717 711 y Fr(0)p Fq(:)p Fr(96)43 b(1)p Fq(:)p Fr(72)717 771 y(2)p Fq(:)p Fr(28)g(0)p Fq(:)p Fr(96)953 668 y Fl(!)1002 741 y Fr(has)17 b(the)f(SVD)342 920 y Fq(A)d Fr(=)h Fq(U)5 b Fr(\006)p Fq(V)557 900 y Fo(T)598 920 y Fr(=)650 847 y Fl( )704 890 y Fr(0)p Fq(:)p Fr(6)42 b Fn(\000)p Fr(0)p Fq(:)p Fr(8)704 950 y(0)p Fq(:)p Fr(8)61 b(0)p Fq(:)p Fr(6)929 847 y Fl(!)9 b( )1024 890 y Fr(3)42 b(0)1024 950 y(0)g(1)1135 847 y Fl(!)8 b( )1230 890 y Fr(0)p Fq(:)p Fr(8)61 b(0)p Fq(:)p Fr(6)1230 950 y(0)p Fq(:)p Fr(6)42 b Fn(\000)p Fr(0)p Fq(:)p Fr(8)1456 847 y Fl(!)1488 859 y Fo(T)-46 1055 y Fr(W)l(e)16 b(see)g(that)h(the)f (columns)f(of)i Fq(U)22 b Fr(and)17 b Fq(V)27 b Fr(are)17 b(unit)f(length)g(since)g(\(0)p Fq(:)p Fr(6\))1312 1037 y Fm(2)1343 1055 y Fr(+)11 b(\(0)p Fq(:)p Fr(8\))1492 1037 y Fm(2)1526 1055 y Fr(=)j(1,)i(and)h(a)g(simple)-46 1115 y(calculation)f(of)g(dot)h(pro)q(ducts)g(will)e(sho)o(w)i(them)e (to)h(b)q(e)h(m)o(utually)d(orthogonal.)27 1175 y(F)l(rom)21 b(the)h(comp)q(onen)o(ts)f(of)i(the)e(SVD,)h(w)o(e)g(can)g(determine)d (man)o(y)i(prop)q(erties)h(of)g(the)g(original)-46 1235 y(matrix.)h(The)18 b Fp(n)n(ull)k(space)17 b Fr(of)h(a)g(matrix)e Fq(A)h Fr(is)g(the)g(set)h(of)f Fq(x)h Fr(for)f(whic)o(h)g Fq(Ax)e Fr(=)h(0,)i(and)g(the)f Fp(range)h Fr(of)-46 1296 y Fq(A)f Fr(is)g(the)g(set)g(of)h Fq(b)f Fr(for)g(whic)o(h)g Fq(Ax)e Fr(=)g Fq(b)i Fr(has)h(a)g(solution)f(for)h Fq(x)p Fr(.)24 b(Let)17 b Fq(u)1275 1303 y Fo(j)1311 1296 y Fr(and)h Fq(v)1431 1303 y Fo(j)1466 1296 y Fr(b)q(e)f(the)g(columns)f (of)i Fq(U)-46 1356 y Fr(and)f Fq(V)27 b Fr(resp)q(ectiv)o(ely)l(.)19 b(Then)d(the)g(decomp)q(osition)g(of)g Fq(A)e Fr(=)g Fq(U)5 b Fr(\006)p Fq(V)1188 1338 y Fo(T)1232 1356 y Fr(can)17 b(b)q(e)f(written)g(as)624 1457 y Fq(Av)685 1464 y Fo(j)717 1457 y Fr(=)d Fq(s)791 1464 y Fo(j)810 1457 y Fq(u)838 1464 y Fo(j)856 1457 y Fq(;)40 b(j)17 b Fr(=)d(1)p Fq(;)8 b Fr(2)p Fq(;)g(:)g(:)g(:)g(;)g(N)r(:)27 1558 y Fr(If)17 b Fq(s)100 1565 y Fo(j)133 1558 y Fr(=)f(0,)h(then)g Fq(Av)415 1565 y Fo(j)448 1558 y Fr(=)e(0)j(and)f Fq(v)662 1565 y Fo(j)697 1558 y Fr(is)g(in)g(the)g(n)o(ull)f(space)h(of)h Fq(A)p Fr(,)e(whereas)h(if)g Fq(s)1496 1565 y Fo(j)1530 1558 y Fn(6)p Fr(=)e(0,)i(then)g Fq(u)1778 1565 y Fo(j)1813 1558 y Fr(is)g(in)-46 1618 y(the)h(range)g(of)h Fq(A)p Fr(.)26 b(Consequen)o(tly)l(,)17 b(w)o(e)h(can)g(construct)g(bases)h (for)f(v)m(arious)h(v)o(ector)e(subspaces)i(de\014ned)-46 1678 y(b)o(y)14 b Fq(A)p Fr(.)20 b(A)14 b(set)h(of)g(v)o(ectors)e Fq(v)458 1685 y Fm(1)478 1678 y Fq(;)8 b(v)524 1685 y Fm(2)543 1678 y Fq(;)g(:)g(:)g(:)f(;)h(v)676 1685 y Fo(k)712 1678 y Fr(in)14 b(a)h(v)o(ector)e(space)i Fq(V)26 b Fr(is)14 b(said)h(to)g(form)e(a)i Fp(basis)h Fr(for)f Fq(V)26 b Fr(if)14 b(ev)o(ery)-46 1739 y(v)o(ector)i Fq(x)h Fr(in)f Fq(V)29 b Fr(can)17 b(b)q(e)g(expressed)f(as)i(a)g(linear)e(com)o (bination)f(of)j(them)d(in)i(exactly)e(one)j(w)o(a)o(y)l(.)23 b(Let)17 b Fq(V)1884 1746 y Fm(0)-46 1799 y Fr(b)q(e)d(the)h(set)f(of)g (columns)g Fq(v)441 1806 y Fo(j)473 1799 y Fr(for)g(whic)o(h)g Fq(s)706 1806 y Fo(j)738 1799 y Fr(=)g(0,)h(and)g(let)e Fq(V)1032 1806 y Fm(1)1067 1799 y Fr(b)q(e)h(the)g(remaining)f(columns) g Fq(v)1651 1806 y Fo(j)1669 1799 y Fr(.)20 b(Similarly)l(,)-46 1859 y(let)d Fq(U)59 1866 y Fm(1)97 1859 y Fr(b)q(e)h(the)g(set)g(of)g (columns)f Fq(u)606 1866 y Fo(j)642 1859 y Fr(for)i(whic)o(h)e Fq(s)883 1866 y Fo(j)918 1859 y Fn(6)p Fr(=)g(0,)h(and)h(let)e Fq(U)1231 1866 y Fm(0)1269 1859 y Fr(b)q(e)h(the)g(remaining)f(columns) f Fq(u)1872 1866 y Fo(j)1890 1859 y Fr(,)-46 1919 y(including)e(those)i (with)e Fq(j)j(>)d(n)p Fr(.)21 b(Th)o(us,)15 b(if)g Fq(k)i Fr(is)d(the)h(n)o(um)o(b)q(er)e(of)j(non-zero)f(singular)h(v)m(alues,)e (there)h(are)g Fq(k)-46 1979 y Fr(columns)g(in)i Fq(V)230 1986 y Fm(0)250 1979 y Fr(,)f Fq(N)h Fn(\000)11 b Fq(k)19 b Fr(columns)d(in)g Fq(V)706 1986 y Fm(1)743 1979 y Fr(and)h Fq(U)871 1986 y Fm(1)891 1979 y Fr(,)g(and)g Fq(M)g Fn(\000)11 b Fq(N)17 b Fr(+)11 b Fq(k)19 b Fr(columns)c(in)i Fq(U)1561 1986 y Fm(0)1581 1979 y Fr(.)22 b(Eac)o(h)17 b(of)g(these)-46 2040 y(sets)f(forms)g(a)g(basis)h(for)g(the)f(v)o(ector)f(subspaces)i (of)g Fq(A)p Fr(.)14 2144 y(1.)24 b Fq(V)104 2151 y Fm(0)140 2144 y Fr(is)16 b(an)h(orthonormal)f(basis)h(for)f Fq(N)5 b(ul)q(l)q(space)p Fr(\()p Fq(A)p Fr(\).)14 2243 y(2.)24 b Fq(V)104 2250 y Fm(1)140 2243 y Fr(is)16 b(an)h(orthonormal)f(basis)h (for)f(the)g(orthogonal)i(complemen)n(t)c(of)i Fq(N)5 b(ul)q(l)q(space)p Fr(\()p Fq(A)p Fr(\).)14 2342 y(3.)24 b Fq(U)109 2349 y Fm(1)145 2342 y Fr(is)16 b(an)h(orthonormal)f(basis)h (for)f Fq(Rang)r(e)p Fr(\()p Fq(A)p Fr(\).)14 2441 y(4.)24 b Fq(U)109 2448 y Fm(0)145 2441 y Fr(is)16 b(an)h(orthonormal)f(basis)h (for)f(the)g(orthogonal)i(compleme)o(n)o(t)13 b(of)k Fq(Rang)r(e)p Fr(\()p Fq(A)p Fr(\).)27 2545 y(As)h(w)o(e)g(shall)g(see) g(in)g(the)h(next)e(t)o(w)o(o)i(sections,)f(the)g(singular)h(v)m(alues) f(of)h Fq(A)f Fr(can)g(b)q(e)h(used)f(in)g(man)o(y)-46 2605 y(other)h(w)o(a)o(ys)f(to)h(determine)d(prop)q(erties)j(of)g Fq(A)p Fr(,)f(as)h(w)o(ell)e(as)j(to)e(partition)h(the)f Fq(M)5 b Fr(-dimensional)18 b(v)o(ector)-46 2666 y(space)e(\(of)h(the)f (mapping)f(de\014ned)h(b)o(y)g Fq(A)p Fr(\))g(in)o(to)g(dominan)o(t)f (and)i(sub-dominan)o(t)f(subspaces.)p eop %%Page: 3 3 3 2 bop -46 -45 a Fu(3)83 b(Prop)r(erties)27 b(of)h(the)f(SVD)-46 80 y Fk(3.1)70 b(SVD)22 b(and)i(Matrix)f(Norms)-46 172 y Fr(Often)d(when)i(sp)q(eaking)f(ab)q(out)h(v)o(ectors)e(and)i (matrices,)e(w)o(e)g(are)h(in)o(terested)f(in)g(the)h(lengths)g(of)g (the)-46 232 y(v)o(ectors)14 b(and)h(the)f(resulting)g(length)g(of)g(a) h(v)o(ector)f(when)g(m)o(ultiplie)o(d)e(b)o(y)i(a)g(matrix.)19 b(A)14 b(familiar)e(concept)-46 292 y(of)h(length)f(in)g(t)o(w)o(o)h (dimensions)e(is)i(the)f(Euclidean)g(distance)g(from)g(the)g(origin)g (to)h(the)g(p)q(oin)o(t)f(sp)q(eci\014ed)g(b)o(y)-46 352 y(the)i(co)q(ordinates)i(of)f(the)f(v)o(ector)g Fn(f)p Fq(x)628 359 y Fm(1)647 352 y Fq(;)8 b(x)697 359 y Fm(2)717 352 y Fn(g)p Fr(.)20 b(This)15 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b(that)g(w)o(e)g(can)h(express)f(a)-46 16 y(matrix)14 b Fq(A)i Fr(as)529 76 y Fq(A)d Fr(=)h Fq(s)654 83 y Fm(1)673 76 y Fq(u)701 83 y Fm(1)721 76 y Fq(v)747 55 y Fo(T)745 88 y Fm(1)785 76 y Fr(+)d Fq(s)857 83 y Fm(2)877 76 y Fq(u)905 83 y Fm(2)924 76 y Fq(v)950 55 y Fo(T)948 88 y Fm(2)988 76 y Fr(+)g Fn(\001)d(\001)g(\001)j Fr(+)g Fq(s)1178 83 y Fo(m)1211 76 y Fq(u)1239 83 y Fo(m)1273 76 y Fq(v)1299 55 y Fo(T)1297 88 y(m)-46 163 y Fr(Eac)o(h)j(outer)h (pro)q(duct)g Fq(u)408 170 y Fo(i)422 163 y Fq(v)448 145 y Fo(T)446 175 y(i)490 163 y Fr(is)f(a)h(simple)d(matrix)h(of)i (rank)f(1,)h(and)g(can)g(b)q(e)g(stored)g(in)f Fq(M)f Fr(+)8 b Fq(N)19 b Fr(n)o(um)o(b)q(ers,)-46 223 y(v)o(ersus)g Fq(M)g Fn(\003)13 b Fq(N)25 b Fr(of)19 b(the)h(original)f(matrix.)29 b(Additionally)l(,)18 b(m)o(ultiplication)f(of)i Fq(u)1482 230 y Fo(i)1496 223 y Fq(v)1522 205 y Fo(T)1520 236 y(i)1569 223 y Fr(with)g(a)h(v)o(ector)e Fq(x)-46 283 y Fr(requires)d(only)h Fq(M)h Fr(+)11 b Fq(N)21 b Fr(op)q(erations,)c(instead)f(of)h Fq(M)f Fn(\003)11 b Fq(N)22 b Fr([FMM77)o(])-46 428 y Fk(3.3)70 b(SVD)22 b(and)i(Linear)f(Indep)r(endence)-46 520 y Fr(Another)16 b(use)h(of)f(the)h(SVD)f(pro)o(vides)g(a)h (measure,)d(called)i(a)g(condition)h(n)o(um)o(b)q(er,)d(whic)o(h)i(is)g (related)g(to)-46 580 y(the)g(measure)f(of)h(linear)g(indep)q(endence)f (b)q(et)o(w)o(een)g(the)h(column)f(v)o(ectors)h(of)g(the)g(matrix.)27 641 y(The)g(condition)g(n)o(um)o(b)q(er)f(\(with)h(resp)q(ect)g(to)h (the)f(Euclidean)f(norm\))g(of)i(a)g(matrix)d Fq(A)i Fr(is)757 762 y Fq(cond)p Fr(\()p Fq(A)p Fr(\))e(=)1001 728 y Fq(s)1024 735 y Fo(max)p 1001 750 95 2 v 1004 796 a Fq(s)1027 803 y Fo(min)-46 892 y Fr(where)h Fq(s)117 899 y Fo(min)198 892 y Fr(and)h Fq(s)315 899 y Fo(max)402 892 y Fr(are)f(the)g(largest)g(and)h(smallest)d(singular)j(v)m(alues)f (of)g Fq(A)p Fr(.)21 b(If)14 b Fq(A)h Fr(is)g(rank)g(de\014cien)o(t,) -46 952 y(then)h Fq(s)88 959 y Fo(i)116 952 y Fr(=)e(0)i(and)h(w)o(e)f (consider)g 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y(system.)591 2114 y Fq(x)619 2121 y Fm(1)704 2114 y Fr(+)54 b(2)p Fq(x)848 2121 y Fm(2)922 2114 y Fr(+)65 b Fq(x)1053 2121 y Fm(3)1139 2114 y Fr(=)h(8)567 2174 y(10)p Fq(x)643 2181 y Fm(1)704 2174 y Fr(+)42 b(18)p Fq(x)860 2181 y Fm(2)922 2174 y Fr(+)f(12)p Fq(x)1077 2181 y Fm(3)1139 2174 y Fr(=)53 b(78)567 2234 y(20)p Fq(x)643 2241 y Fm(1)704 2234 y Fr(+)42 b(22)p Fq(x)860 2241 y Fm(2)922 2234 y Fr(+)f(40)p Fq(x)1077 2241 y Fm(3)1139 2234 y Fr(=)g(144)27 2318 y(This)15 b(problem)e(can)i(b)q(e)f(expressed)g(in)h(terms)e(of)i (a)g(co)q(e\016cien)o(t)e(matrix)g Fq(A)p Fr(,)h(a)h(v)o(ector)e Fq(x)i Fr(of)g(v)m(ariables,)-46 2378 y(and)20 b(a)h(v)o(ector)d Fq(b)p Fr(,)i(suc)o(h)g(that)g(a)h(solution)f(to)g(the)f(linear)g (system)g Fq(Ax)g Fr(=)h Fq(b)f Fr(is)h(an)g(assignmen)o(t)f(to)h(the) -46 2438 y(v)m(alues)c(of)h(the)f(v)o(ector)f Fq(x)p Fr(.)21 b(F)l(or)16 b(the)g(ab)q(o)o(v)o(e)h(example,)c Fq(A;)8 b(x)p Fr(,)15 b(and)i Fq(b)f Fr(are)438 2604 y Fq(Ax)d Fr(=)568 2506 y Fl(0)568 2580 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Fo(j)504 615 y Fr(can)h(b)q(e)g(set)f(to)h(an)o(y)f(v)m (alue)g(and)h(still)f(yield)f(a)i(solution.)25 b(A)17 b(general)g(form)-46 675 y(of)g(the)g(p)q(ossible)g(solutions)g(can)g (then)g(b)q(e)g(expressed)f(in)g(terms)f(of)j(these)e(arbitrary)h(comp) q(onen)o(ts)f(of)h Fq(z)-46 735 y Fr(when)f(transformed)g(bac)o(k)g(to) g(the)g(original)g(co)q(ordinates)i(b)o(y)d Fq(x)f Fr(=)g Fq(V)d(z)r Fr(.)27 795 y(The)17 b(condition)f(n)o(um)o(b)q(er)f(of)i(a) g(matrix)d(can)j(also)g(describ)q(e)f(the)g(sensitivit)o(y)f(of)i (solutions)g(of)g(linear)-46 855 y(systems)g(to)h(inaccuracies)f(in)g (the)h(data.)27 b(Supp)q(ose)19 b(w)o(e)f(w)o(an)o(t)g(to)g(measure)f (the)g(maximal)e(increase)i(in)-46 915 y(relativ)o(e)c(inaccuracy)h (for)i(the)e(w)o(orst)i(p)q(osition)g(of)f Fq(b)g Fr(and)g(error)g Fq(db)p Fr(,)g(when)g(solving)g(for)g Fq(x)g Fr(in)g(the)f(system)-46 976 y Fq(Ax)f Fr(=)h Fq(b)p Fr(.)21 b(The)16 b(answ)o(er)h(is)f (precisely)e(the)i(condition)g(n)o(um)o(b)q(er)f([VDM88)o(].)474 1109 y Fq(E)h Fr(=)e Fq(cond)p Fr(\()p Fq(A)p Fr(\))g(=)g(max)833 1139 y Fo(b;db)921 1076 y Fn(k)g Fq(dx)g Fn(k)g Fq(=)g Fn(k)g Fq(x)f Fn(k)p 921 1098 288 2 v 923 1143 a(k)h Fq(dy)i Fn(k)e Fq(=)g Fn(k)g Fq(y)h Fn(k)1228 1109 y Fr(=)1285 1076 y Fq(s)1308 1083 y Fo(max)p 1285 1098 95 2 v 1288 1143 a Fq(s)1311 1150 y Fo(min)-46 1241 y Fr(F)l(or)h(the)g(ab)q(o)o(v)o(e)h(reason,)f(matrices)e(with)j(large)f (condition)g(n)o(um)o(b)q(ers)f(are)h(said)g(to)h(b)q(e)f (ill-conditioned.)27 1301 y(As)h(an)h(extension)f(of)h(solving)f (linear)g(systems,)f(supp)q(ose)i(w)o(e)f(wish)h(to)f(\014nd)h(a)g (solution)g(where)f Fq(Ax)-46 1361 y Fr(is)f(appro)o(ximately)e(equal)i (to)g Fq(b)p Fr(.)21 b(By)16 b(this)g(w)o(e)g(mean)f(the)h Fp(least-squares)g Fr(solution)g Fq(x)g Fr(to)h(minim)o(ize)801 1467 y Fn(k)c Fq(Ax)e Fn(\000)g Fq(b)i Fn(k)1024 1447 y Fm(2)1044 1467 y Fq(;)-46 1573 y Fr(or)j(equiv)m(alen)o(tly)e(to)i (minim)o(iz)o(e)d(the)i(length)h Fn(k)e Fq(Ax)9 b Fn(\000)h Fq(b)j Fn(k)p Fr(.)21 b(The)16 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b(if)16 b Fq(s)1077 2153 y Fo(j)1109 2146 y Fr(=)d(0)-46 2249 y(Hence,)h Fq(k)j Fr(of)f(the)g(equations)f(ha)o(v)o(e)g(exact)g(solutions)h(and)g(the)f (remaining)f(ones)i(yield)e(a)i(p)q(ossibly)g(non-)-46 2309 y(zero)f(residual)g(v)o(ector)g(of)g(length)g(\()604 2276 y Fl(P)657 2309 y Fq(d)682 2291 y Fm(2)682 2321 y Fo(i)702 2309 y Fr(\))726 2277 y Fj(1)p 726 2283 16 2 v 726 2304 a(2)748 2309 y Fr(,)g(where)g(the)g(sum)g(is)g(o)o(v)o(er) f(all)h(i)g(for)h(whic)o(h)f Fq(s)1586 2316 y Fo(i)1614 2309 y Fr(=)e(0)j(or)g Fq(i)d(>)h(N)5 b Fr(.)-46 2369 y(The)16 b(solution)h(to)f(the)g(original)h(problem)d(is)i(then)g Fq(x)e Fr(=)g Fq(V)d(z)18 b Fr([FMM77].)-46 2513 y Fk(4.2)70 b(Noisy)22 b(Signal)g(Filtering)-46 2605 y Fr(Problems)14 b(in)g(signal)i(pro)q(cessing)f(often)h(use)f(linear)f(mo)q(dels)g(for) h(signals.)21 b(In)15 b(ideal)f(\(noise-free\))h(condi-)-46 2666 y(tions,)j(the)f(measurmen)o(t)e(data)j(can)g(b)q(e)f(arranged)i (in)e(a)h(matrix,)e(where)h(the)g(matrix)f(is)h(kno)o(wn)h(to)g(b)q(e)p eop %%Page: 7 7 7 6 bop -46 -45 a Fr(rank-de\014cien)o(t.)27 b(By)18 b(this,)h(w)o(e)f(mean)g(that)h(the)f(signal)h(is)f(assumed)g(to)h(lie) f(in)g(a)h(prop)q(er)h(subspace)f(of)-46 16 y(Euclidean)e(space.)27 b(Ho)o(w)o(ev)o(er,)16 b(the)h(presence)g(of)i(noise,)e(either)g(from)g (rounding)i(error)f(or)g(instrumen)o(t)-46 76 y(error,)c(results)g(in)g (a)h(measuremen)o(t)c(matrix)h(that)j(is)f(often)g(of)h(full)e(rank.)21 b(Usually)l(,)13 b(the)h(mo)q(dels)g(assume)-46 136 y(that)19 b(the)g(error)g(can)h(b)q(e)f(separated)g(from)f(the)h(data,)h(in)f (that)g(the)g(noise)g(comp)q(onen)o(t)f(is)h(that)h(whic)o(h)-46 196 y(lies)f(in)g(a)h(subspace)g(orthogonal)i(to)e(the)f(signal)h (subspace.)32 b(F)l(or)20 b(this)f(reason,)i(the)f(SVD)f(is)h(used)g (to)-46 256 y(appro)o(ximate)d(the)h(matrix,)f(decomp)q(osing)i(the)f (data)h(in)o(to)g(an)g(optimal)e(estimate)f(of)j(the)g(signal)f(and)-46 317 y(the)e(noise)g(comp)q(onen)o(ts.)27 377 y(Supp)q(ose)e Fq(A)f Fr(is)g(the)g(measuremen)o(t)d(matrix,)i(where)h(eac)o(h)g (column)f(consists)h(of)h(a)f(signal)h(comp)q(onen)o(t)-46 437 y Fq(x)i Fr(and)h(a)f(noise)h(comp)q(onen)o(t)e Fq(n)p Fr(.)679 497 y Fq(A)e Fr(=)h(\()p Fq(C)835 504 y Fm(1)868 497 y Fn(j)g Fq(C)931 504 y Fm(2)964 497 y Fn(j)g(\001)8 b(\001)g(\001)14 b(j)g Fq(C)1127 504 y Fo(N)1160 497 y Fr(\))-46 582 y(where)k(eac)o(h)g Fq(C)244 589 y Fo(i)276 582 y Fr(=)f Fq(x)359 589 y Fo(i)386 582 y Fr(+)12 b Fq(n)465 589 y Fo(i)479 582 y Fr(.)28 b(The)19 b(v)o(ector)e Fq(x)i Fr(represen)o(ting)e(the)h(signal)h(is)f(kno)o(wn)h(to)g(lie)e (in)h(a)h(rank)g Fq(k)-46 643 y Fr(subspace,)e(though)g(the)f(precise)g (subspace)h(is)f(not)h(kno)o(wn.)22 b(Therefore,)16 b(let)f Fq(x)f Fr(=)g Fq(H)t(c)j Fr(for)g(a)g(co)q(e\016cien)o(t)-46 703 y(v)o(ector)e Fq(c)g Fr(and)h(a)g(matrix)e Fq(H)19 b Fr(whose)d(columns)f(are)g(the)g(basis)h(v)o(ectors)f(of)h(some)e (rank)i Fq(k)h Fr(subspace.)22 b(The)-46 763 y(\(least-squared\))13 b(error)f(b)q(et)o(w)o(een)g Fq(A)g Fr(and)h Fq(H)t(c)g Fr(is)g(minim)o(iz)o(ed)d(b)o(y)i(c)o(ho)q(osing)h Fq(H)k Fr(to)c(b)q(e)g(the)f(optimal)f Fq(k)k Fr(rank)-46 823 y(appro)o(ximation)i Fq(A)318 830 y Fo(k)356 823 y Fr(to)h Fq(A)p Fr(.)25 b(Then)18 b(the)g Fq(k)h Fr(columns)e(of)h Fq(U)5 b Fr(,)18 b(corresp)q(onding)h(to)f(the)f Fq(k)j Fr(largest)e(singular)-46 883 y(v)m(alues,)e(span)h(the)f(rank)g Fq(k)i Fr(subspace)f Fq(H)t Fr(.)k(The)16 b(resulting)g(error)g(is)g Fq(e)1241 865 y Fm(2)1274 883 y Fr(=)1326 850 y Fl(P)1370 863 y Fo(N)1370 896 y(k)q Fm(+1)1445 883 y Fq(s)1468 865 y Fm(2)1468 896 y Fo(i)1487 883 y Fr(.)21 b(Using)16 b(the)g(SVD)g(as)-46 943 y(ab)q(o)o(v)o(e,)d(w)o(e)g(see)g(that)h(the)f (original)g(data)h(matrix)d Fq(A)i Fr(is)g(decomp)q(osed)g(in)o(to)g (the)g(orthogonal)i(comp)q(onen)o(ts)-46 1004 y Fq(U)5 b Fr(\006)27 1011 y Fo(k)49 1004 y Fq(V)88 986 y Fo(T)116 1004 y Fr(,)15 b(whic)o(h)g(is)h(the)f(rank)i Fq(k)g 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b(then)g(\014ltered)g(using)g(the)g(Discrete)f(F)l (ourier)h(T)l(ransform)g(to)h(remo)o(v)o(e)c(signal)k(noise.)27 b(Eac)o(h)18 b(v)o(ector)-46 1601 y Fq(b)i Fr(=)f Fn(f)p Fq(b)98 1608 y Fm(1)118 1601 y Fq(;)8 b(b)161 1608 y Fm(2)180 1601 y Fq(;)g(:)g(:)g(:)f(;)h(b)310 1608 y Fo(M)349 1601 y Fn(g)20 b Fr(represen)o(ts)f(a)h(state)g(of)g(the)g(underlying)f (dynamical)f(system.)30 b(The)20 b(ob)s(ject)f(is)-46 1661 y(to)i(\014nd)g(the)f(b)q(est)h(\(least-squares)g(distance\))g Fq(L)p Fr(-dimensional)e(linear)h(space)h(\()p Fq(L)g Fn(\024)g Fq(M)5 b Fr(\))21 b(that)g(passes)-46 1721 y(through)h(the)f(cen)o(ter)e(of)j(mass)e Fq(c)h Fr(of)g(the)g Fq(K)k Fr(nearest)c(neigh)o(b)q(ors)h(of)f Fq(b)p Fr(.)35 b(W)l(e)21 b(construct)g(a)g(matrix)e Fq(A)-46 1781 y Fr(whose)h(ro)o(ws)f(consist)h(of)f(the)g(v)o(ectors)g Fq(b)715 1788 y Fm(1)747 1781 y Fn(\000)13 b Fq(c;)8 b(b)863 1788 y Fm(2)896 1781 y Fn(\000)k Fq(c;)c(:)g(:)g(:)g(;)g(b)1099 1788 y Fo(k)1133 1781 y Fn(\000)13 b Fq(c)p Fr(,)19 b(and)h(calculate)e (the)h(SVD)g Fq(U)5 b Fr(\006)p Fq(V)1863 1763 y Fo(T)1890 1781 y Fr(.)-46 1842 y(T)l(aking)21 b(the)g(\014rst)g Fq(L)g Fr(columns)e(of)i(the)g(orthogonal)h(matrix)d Fq(V)32 b Fr(giv)o(es)20 b(us)h(the)g(desired)f(basis)h(for)g(the)-46 1902 y Fq(L)p Fr(-dimensional)15 b(subspace.)473 1884 y Fm(1)-46 2068 y Fu(5)83 b(Discussion)25 b(and)j(Conclusion)-46 2177 y Fr(By)13 b(pro)o(viding)h(an)g(appro)o(ximation)f(to)i(rank)f (de\014cien)o(t)e(matrices,)g(and)j(exp)q(osing)f(the)g(geometric)e (prop-)-46 2237 y(erties)19 b(of)g(the)h(matrix,)e(the)h(singular)h(v)m (alue)f(decomp)q(osition)g(of)h(a)g(matrix)d(is)j(a)g(p)q(o)o(w)o (erful)f(tec)o(hnique)-46 2297 y(in)e(matrix)e(computations.)23 b(Despite)17 b(its)g(usefulness,)g(ho)o(w)o(ev)o(er,)e(there)i(are)g(a) h(n)o(um)o(b)q(er)d(of)i(dra)o(wbac)o(ks,)-46 2358 y(as)j(men)o(tioned) d(b)o(y)h([V)l(ac91].)30 b(F)l(or)19 b(problems)f(that)h(can)g(b)q(e)h (solv)o(ed)e(b)o(y)h(simpler)e(tec)o(hniques,)g(suc)o(h)i(as)-46 2418 y(the)j(F)l(ourier)g(T)l(ransform,)h(or)f Fq(QR)g Fr(decomp)q(osition,)h(use)f(of)h(the)f(SVD)g(ma)o(y)e(b)q(e)j(unduly)e (exp)q(ensiv)o(e)-46 2478 y(computationally)l(.)e(Secondly)l(,)14 b(the)h(SVD)g(op)q(erates)h(on)g(a)f(\014xed)g(matrix,)e(hence)i(it)f (is)h(not)h(amenable)e(to)p -46 2520 780 2 v 10 2551 a Ff(1)29 2566 y Fe(The)g(\014rst)h Fd(L)f Fe(columns)f(of)g Fd(V)23 b Fe(pro)o(vide)14 b(a)g(basis)g(for)f Fd(N)5 b(ul)q(l)q(S)r(P)h(ace)p Fe(\()p Fd(A)p Fe(\))13 b(=)f Fd(Rang)q(e)p Fe(\()p Fd(A)1339 2551 y Fc(T)1366 2566 y Fe(\))i(whic)o(h)f(is)h(consisten)o(t)h(with)f(our)-46 2616 y(previous)f(discussion)h(since)g(the)f(v)o(ectors)h Fd(b)641 2622 y Fc(i)662 2616 y Fb(\000)8 b Fd(c)13 b Fe(constitute)h(the)f Fa(r)n(ows)f Fe(of)h Fd(A)p Fe(,)g(rather)g(than) g(the)h Fa(c)n(olumns)f Fe(as)g(in)g(the)g(other)-46 2666 y(examples.)p eop %%Page: 8 8 8 7 bop -46 -45 a Fr(problems)18 b(that)h(require)f(adaptiv)o(e)g (algorithms.)29 b(A)19 b(host)g(of)h(activ)o(e)d(researc)o(h)i (e\013orts)h(address)f(these)-46 16 y(problems.)25 b(F)l(urther)18 b(examples)e(of)i(the)g(use)g(of)h(SVD)f(in)f(the)h(\014eld)g(of)g (Signal)g(Pro)q(cessing,)h(as)g(w)o(ell)d(as)-46 76 y(discussions)d(of) f(implem)o(en)o(tation)e(algorithms)h(and)i(arc)o(hitectures,)e(can)i (b)q(e)f(found)h(in)f([V)l(ac91,)g(Dep88)q(].)-46 242 y Fu(References)-16 352 y Fr([And86])24 b(F)l(ob)q(ert)g(F.)f(V.)g (Anderson.)43 b Fh(Intr)n(o)n(duction)23 b(to)i(Line)n(ar)e(A)o(lgebr)n (a)p Fr(.)44 b(Holt,)24 b(Rinehart,)g(and)175 412 y(Winston,)17 b(New)e(Y)l(ork,)h(1986.)1 514 y([DD88])24 b(P)l(.)12 b(Dewilde)g(and)h(Ed.)f(F.)f(Deprettere.)j(Singular)e(v)m(alue)g (decomp)q(osition:)19 b(An)12 b(in)o(tro)q(duction.)175 574 y(In)21 b(Ed.)f(F.)g(Deprettere,)g(editor,)h Fh(SVD)h(and)f(Signal) i(Pr)n(o)n(c)n(essing:)30 b(A)o(lgorithms,)22 b(Applic)n(a-)175 634 y(tions,)d(and)f(A)o(r)n(chite)n(ctur)n(es)p Fr(,)e(pages)h(3{41.)h (Elsevier)d(Science)g(Publishers,)h(North)h(Holland,)175 694 y(1988.)-11 796 y([Dep88])24 b(Ed.)16 b(F.)e(Deprettere,)g(editor.) 20 b Fh(SVD)d(and)g(Signal)h(Pr)n(o)n(c)n(essing:)j(A)o(lgorithms,)c(A) o(nalysis,)g(and)175 856 y(Applic)n(ations)p Fr(.)22 b(Elsevier)15 b(Science)g(Publishers,)g(North)i(Holland,)e(1988.)-46 958 y([FMM77])23 b(George)15 b(E.)e(F)l(orsythe,)h(Mic)o(hael)e(A.)h (Malcolm,)f(and)i(Clev)o(e)f(B.)g(Moler.)j Fh(Computer)g(Metho)n(ds)175 1018 y(for)h(Mathematic)n(al)h(Computations)p Fr(,)f(pages)g(201{235.) 24 b(Pren)o(tice)15 b(Hall,)h(Englew)o(o)q(o)q(d)h(Cli\013s,)175 1078 y(1977.)-30 1180 y([GVL83])24 b(Gene)d(H.)f(Golub)i(and)f(Charles) g(F.)g(V)l(an)g(Loan.)36 b Fh(Matrix)22 b(Computations)p Fr(,)g(pages)f(16{21,)175 1240 y(293.)i(Johns)17 b(Hopkins)f(Univ)o (ersit)o(y)d(Press,)k(Baltimore,)c(Maryland,)j(1983.)-3 1342 y([Sau94])24 b(Tim)19 b(Sauer.)35 b(Time)18 b(series)i(prediction) g(b)o(y)g(using)h(dela)o(y)o(ed)e(co)q(ordinate)i(em)o(b)q(edding.)32 b(In)175 1402 y(Andreas)22 b(S.)g(W)l(eigend)f(and)i(Neil)d(A.)h (Gershenfeld,)h(editors,)h Fh(Time)g(Series)g(Pr)n(e)n(diction:)175 1462 y(F)l(or)n(e)n(c)n(asting)18 b(the)f(F)l(utur)n(e)h(and)g (Understanding)h(the)f(Past)p Fr(.)d(Adgdison-W)l(esley)l(,)h(1994.)1 1564 y([Sc)o(h91])23 b(Louis)f(L.)e(Sc)o(harf.)33 b(The)21 b(SVD)f(and)h(reduced-rank)f(signal)h(pro)q(cessing.)34 b(In)21 b(R.)e(V)l(accaro,)175 1624 y(editor,)12 b Fh(SVD)i(and)f (Signal)i(Pr)n(o)n(c)n(essing)e(II:)g(A)o(lgorithms,)h(Applic)n (ations,)g(and)g(A)o(r)n(chite)n(ctur)n(es)p Fr(,)175 1684 y(pages)j(3{31.)h(Elsevier)d(Science)f(Publishers,)i(North)g (Holland,)g(1991.)-3 1786 y([V)l(ac91])23 b(R.)g(V)l(accaro,)i(editor.) 42 b Fh(SVD)25 b(and)f(Signal)h(Pr)n(o)n(c)n(essing)f(II:)f(A)o (lgorithms,)j(A)o(nalysis,)g(and)175 1846 y(Applic)n(ations)p Fr(.)c(Elsevier)15 b(Science)g(Publishers,)g(North)i(Holland,)e(1991.) -43 1948 y([VDM88])23 b(Jo)q(os)e(V)l(andew)o(alle)d(and)h(Bart)g(De)g (Mo)q(or.)31 b(A)18 b(v)m(ariet)o(y)g(of)i(applications)f(of)g (singular)g(v)m(alue)175 2008 y(decomp)q(osition)12 b(in)g(iden)o (ti\014cation)g(and)h(signal)g(pro)q(cessing.)i(In)e(Ed.)f(F.)g (Deprettere,)g(editor,)175 2068 y Fh(SVD)22 b(and)f(Signal)i(Pr)n(o)n (c)n(essing:)29 b(A)o(lgorithms,)22 b(Applic)n(ations,)h(and)e(A)o(r)n (chite)n(ctur)n(es)p Fr(,)f(pages)175 2128 y(43{91.)e(Elsevier)d (Science)g(Publishers,)g(North)h(Holland,)g(1988.)-12 2230 y([W)o(G94])24 b(Andreas)14 b(S.)e(W)l(eigend)h(and)g(Neil)f(A.)g (Gershenfeld.)j Fh(Time)g(Series)g(Pr)n(e)n(diction:)20 b(F)l(or)n(e)n(c)n(asting)175 2290 y(the)c(F)l(utur)n(e)e(and)h (Understanding)h(the)f(Past)p Fr(.)h(Addison-W)l(esley)l(,)d(Reading,)g (Massac)o(h)o(usetts,)175 2351 y(1994.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF