- 1. SUMMER PROJECT REPORT RAVINDER SINGH, IMS13114 NAME : Ravinder Singh, ID: IMS13114. GUIDE : Dr. Sachindranath Jayaraman, IISER-TVM. I reported as a Summer Project Student to Dr. Sachindranath Jayaraman, IISER- TVM, on May 1,2015. I started reading the book Introduction to Linear Algebra, by Prof.Gilbert Strang and I also watched all the video lectures of the online course Linear Algebra 18.06, available at ocw.mit.edu. Then I started reading the paper Down with determinants!, by Sheldon Axler, published in the American Mathematical Monthly [1]. This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure theorems of linear algebra without resorting to determinants. A brief summary of the important results I learnt from the above paper as well as from other references are described below. Deﬁnition 1.1. Given an m ∗ n matrix A having columns A1, A2, ....An and rows a1, a2, ....am, let there be another matrix B of size n∗p and have columns B1, B2, ....Bp and rows b1, b2, ....bn and a matrix C = A∗B which has columns C1, C2, ....Cp and rows c1, c2, ....cm.Now the matrix C can be obtained in the following ways. (a) Ci = A ∗ Bi ∀ i ∈ {1, 2, ....p} (b) cj = aj ∗ B ∀ j ∈ {1, 2, ....m} (c) Cij = ai ∗ Bj ∀ i ∈ {1, 2, ....m} and ∀ j ∈ {1, 2, ....p} (d) C = i=1,2,....n (Ai ∗ bi) Lemma 1.2. Steinitz Exchange Lemma : If {v1, v2, ....vm} is a set of m linearly independent vectors in a vector space V , and {w1, w2, ....wn} span V then m n and, possibly after reordering the wi, the set {v1, v2, ....vm, wm+1, wm+2, ....wn} spans V . Then using the Steinitz Exchange Lemma , I proved the following theorem. Deﬁnition 1.3. Given an m ∗ n matrix A, its Column rank (rc) is deﬁned as the num- ber of independent columns of matrix A and similarly its Row rank (rr) is deﬁned as 1
- 2. 2 RAVINDER SINGH, IMS13114 the number of independent rows of matrix A. Theorem 1.4. Given an m ∗ n matrix A.Then rc = rr. Theorem 1.5. Given an m ∗ n matrix A having rank = r.Then A ∗ x = b has solution as follows: (a) If r = m = n , then there exists 1 unique solution. (b) If r = n < m , then there exists 0 or 1 solutions. (c) If r = m < n , then there exists ∞ solutions. (d) If r < m and r < n , then there exists 0 or ∞ solutions. After this, I learned 4 diﬀerent types of matrix decompositions. Given an m ∗ n matrix A having rank r, then followings are the descriptions of its diﬀerent types of decompositions. (a) A = L ∗ U, where L is a lower triangular matrix and U is an upper triangular matrix. (b) A = B ∗C, where B is a full column rank matrix and C is the full row rank matrix. (c) A = Q ∗ R, where Q is an orthogonal matrix and R is an upper triangular matrix. (d) A = S ∗ Λ ∗ S−1, where S is an eigenvector matrix and Λ is the diagonal matrix with eigenvalues as its diagonal entries. Theorem 1.6. Fundamental Theorem of Linear Algebra : Given an m ∗ n matrix A, the fundamental theorem of linear algebra is a collection of results relating various properties of the four fundamental matrix subspaces of A. In particular: (a) dim(R(A)) = dim(R(AT )) and dim(R(A)) + dim(N(A)) = n where here, R(A) denotes the range or column space of A, AT denotes its transpose, and N(A) denotes its null space. (b) The null space N(A) is orthogonal to the row space R(AT ). After this, I proved the Euler s Formula for graphs, which is stated below: Given a graph with n number of nodes, e number of edges, l number of loops, then n − e + l = 1. I also learned that for a matrix A to have a set of orthogonal eigenvectors, it should have a property that A ∗ AT = AT ∗ A.
- 3. SUMMER PROJECT REPORT 3 Now, I started reading the paper Down with determinants!, by Sheldon Axler, pub- lished in the American Mathematical Monthly [1]. Deﬁnition 1.7. A complex number λ is called an eigenvalue of a linear operator T if T − λ ∗ I is not injective. Theorem 1.8. Every linear operator on a ﬁnite-dimensional complex vector space has an eigenvalue. Proposition 1.9. Non-zero eigenvectors corresponding to distinct eigenvalues of T are linearly independent. Deﬁnition 1.10. A vector v ∈ V is called a generalisedeigenvector of linear operator T if (T − λ ∗ I)k = 0, for some eigenvalue λ of T and some positive integer k. Lemma 1.11. The set of generalised eigenvectors of T corresponding to an eigenvalue λ equals ker(T − λ ∗ I)n, where n is the dimension of the V . Proposition 1.12. The generalised eigenvectors of T span V . Proposition 1.13. Non-zero generalised eigenvectors corresponding to distinct eigen- values of T are linearly independent. Theorem 1.14. Let λ1, ....., λm be the distinct eigenvalues of T, with U1,.....,Um de- noting the corresponding sets of generalized eigenvectors. Then (a) V =U1 ⊕ · · · ⊕ Um; (b) T maps each Uj into itself; (c) each (T − λjI)|Uj is nilpotent; (d) each T|Uj has only one eigenvalue, namely λj. Deﬁnition 1.15. The Minimal Polynomial : There is a smallest positive integer k such that I, T, T2, ...., Tk are not linearly independent. Thus there exist unique complex numbers a0, a1, ....ak−1 such that a0 ∗I +a1 ∗T +a2 ∗T2 +....ak−1 ∗Tk−1 +Tk = 0. The polynomial a0+a1∗z+a2∗z2+....ak−1∗zk−1+zk is called the minimal polynomial of T.
- 4. 4 RAVINDER SINGH, IMS13114 Theorem 1.16. Let λ1, λ2, ....λm be the distinct eigenvalues of T, let Uj denote the set of generalised eigenvectors corresponding to λj, and let αj be the smallest positive inte- ger such that (T −λj ∗I)αj ∗v = 0 for every v ∈ Uj. Let p(z) = (z−λ1)α1 ....(z−λm)αm . Then (a) p is the minimal polynomial of T; (b) p has degree at most dimV ; (c) if q is a polynomial such that q(T) = 0, then q is a polynomial multiple of p. Deﬁnition 1.17. Let λ1, λ2, ....λm be the distinct eigenvalues of T, with correspond- ing multiplicities β1, β2, ....βm..The polynomial (z − λ1)β1 ....(z − λm)βm is called the characteristic polynomial of T. Theorem 1.18. Let q denote the characteristic polynomial of T. Then q(T) = 0. Lemma 1.19. Suppose T is nilpotent. Then there is a basis of V with respect to which the matrix of T contains only 0’s on and below the main diagonal. Theorem 1.20. Let λ1, λ2, ....λm be the distinct eigenvalues of T. Then there is a basis of V with respect to which the matrix of T has the form M = λ1 ∗ 0 0 0 0 0 0 0 0 0 . ∗ 0 0 0 0 0 0 0 0 0 . ∗ 0 0 0 0 0 0 0 0 0 λ1 ∗ 0 0 0 0 0 0 0 0 0 . ∗ 0 0 0 0 0 0 0 0 0 . ∗ 0 0 0 0 0 0 0 0 0 λm ∗ 0 0 0 0 0 0 0 0 0 . ∗ 0 0 0 0 0 0 0 0 0 . ∗ 0 0 0 0 0 0 0 0 0 λm Lemma 1.21. If T is normal, then kerT = kerT∗. Proposition 1.22. Every generalised eigenvector of a normal operator is an eigenvec- tor of the operator.
- 5. SUMMER PROJECT REPORT 5 Proposition 1.23. Eigenvectors of a normal operator corresponding to distinct eigen- values are orthogonal. Theorem 1.24. There is an orthonormal basis of V consisting of eigenvectors of T if and only if T is normal. Proposition 1.25. Every eigenvalue of a self-adjoint operator is real. Theorem 1.26. Every linear operator on an old-dimensional real vector space has a real eigenvalue. Theorem 1.27. Suppose U is a real inner product space and S is a linear operator on U. Then there is an orthonormal basis of U consisting of eigenvectors of S if and only if S is self-adjoint. Theorem 1.28. An operator is invertible if and only if its determinant is non-zero. Proposition 1.29. The characteristic polynomial of T equals det(z ∗ I − T). The classical deﬁnition of determinant goes as follows: The determinant of T is deﬁned as det(T) = π∈Sn (signπ)tπ(1),1 . . . tπ(n),n, where Sn denotes the permutation group on n symbols. This is less intuitive at ﬁrst. However, a nice and simple geomet- ric meaning of the determinant is through the change of variable formula, the statement of which goes as follows. Lemma 1.30. Let S be a linear operator on a real inner product space U. Then there exists a linear isometry A on U such that S = A ∗ √ S∗S. Theorem 1.31. Let S be a linear operator on Rn. Then vol S(E) = det(S) ∗ volE for E ⊂ Rn. Theorem 1.32. The Schur decomposition reads as follows: if A is a n ∗ n square ma- trix with complex entries, then A can be expressed as A = Q ∗ U ∗ Q−1, where Q is a unitary matrix (so that its inverse Q−1 is also the conjugate transpose Q∗ of Q), and
- 6. 6 RAVINDER SINGH, IMS13114 U is an upper triangular matrix, which is called a Schurform of A. Since U is similar to A, it has the same multiset of eigenvalues, and since it is trian- gular, those eigenvalues are the diagonal entries of U. Theorem 1.33. Matrix polar decomposition : The polar decomposition of a square complex matrix A is a matrix decomposition of the form A = U ∗ P, where U is a unitary matrix and P is a positive-semideﬁnite Hermitian matrix. Intuitively, the polar decomposition separates A into a component that stretches the space along a set of orthogonal axes, represented by P, and a rotation (with possible reﬂection) represented by U. The decomposition of the complex conjugate of A is given by A = U * P. This decomposition always exists; and so long as A is invertible, it is unique, with P positive-deﬁnite. Theorem 1.34. Singular V alue Decomposition : Suppose M is a m∗n matrix whose entries come from the ﬁeld K, which is either the ﬁeld of real numbers or the ﬁeld of complex numbers. Then there exists a factorization of the form M = U ∗ Σ ∗ V ∗, where U is an m m unitary matrix over K (orthogonal matrix if K = R), is a m∗n diagonal matrix with non-negative real numbers on the diagonal, and the n ∗ n unitary matrix V ∗ denotes the conjugate transpose of the n ∗ n unitary matrix V . Such a factorization is called a singular value decomposition of M. The diagonal entries σi of Σ are known as the singular values of M. A common convention is to list the singular values in descending order. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not). Theorem 1.35. If E is a subset of Rn and S is a linear transformation on Rn, then the volume of S(E) must equal the volume of E multiplied by the absolute value of the product of the eigenvalues of S, counting multiplicity.
- 7. SUMMER PROJECT REPORT 7 References 1. S. Axler, Down with determinants!, The American Mathematical Monthly, Vol. 102(2)(1995), 139- 154. 2. S. Axler, Linear Algebra Done Right, 3rd edition, Springer verlag, 2010. 3. Gilbert Strang, Introduction to Linear Algebra, 4th edition, Wellesley Cambridge Press, 2009. 4. Gilbert Strang, Linear Algebra 18.06, ocw.mit.edu(Video Lectures), fall 1999. Indian Institute of Science Education and Research,Thiruvananthapuram-695016,Kerala,India E-mail address: ravi-singh5851113@iisertvm.ac.in