DOCSLIB.ORG

Math 2331 – Linear Algebra 6.2 Orthogonal Sets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 2331 – Linear Algebra 6.2 Orthogonal Sets 6.2 Orthogonal Sets Math 2331 { Linear Algebra 6.2 Orthogonal Sets Jiwen He Department of Mathematics, University of Houston [email protected] math.uh.edu/∼jiwenhe/math2331 Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix 6.2 Orthogonal Sets Orthogonal Sets: Examples Orthogonal Sets: Theorem Orthogonal Basis: Examples Orthogonal Basis: Theorem Orthogonal Projections Orthonormal Sets Orthonormal Matrix: Examples Orthonormal Matrix: Theorems Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets Orthogonal Sets n A set of vectors fu1; u2;:::; upg in R is called an orthogonal set if ui · uj = 0 whenever i 6= j. Example 82 3 2 3 2 39 < 1 1 0 = Is 4 −1 5 ; 4 1 5 ; 4 0 5 an orthogonal set? : 0 0 1 ; Solution: Label the vectors u1; u2; and u3 respectively. Then u1 · u2 = u1 · u3 = u2 · u3 = Therefore, fu1; u2; u3g is an orthogonal set. Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Sets: Theorem Theorem (4) Suppose S = fu1; u2;:::; upg is an orthogonal set of nonzero n vectors in R and W =spanfu1; u2;:::; upg. Then S is a linearly independent set and is therefore a basis for W . Partial Proof: Suppose c1u1 + c2u2 + ··· + cpup = 0 (c1u1 + c2u2 + ··· + cpup) · = 0· (c1u1) · u1 + (c2u2) · u1 + ··· + (cpup) · u1 = 0 c1 (u1 · u1) + c2 (u2 · u1) + ··· + cp (up · u1) = 0 c1 (u1 · u1) = 0 Since u1 6= 0, u1 · u1 > 0 which means c1 = : In a similar manner, c2,:::,cp can be shown to by all 0. So S is a linearly independent set. Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis Orthogonal Basis: Example An orthogonal basis for a subspace W of Rn is a basis for W that is also an orthogonal set. Example Suppose S = fu1; u2;:::; upg is an orthogonal basis for a subspace n W of R and suppose y is in W . Find c1,:::,cp so that y =c1u1 + c2u2 + ··· + cpup. Solution: y· = (c1u1 + c2u2 + ··· + cpup) · y · u1= (c1u1 + c2u2 + ··· + cpup) · u1 y · u1=c1 (u1 · u1) + c2 (u2 · u1) + ··· + cp (up · u1) y · u =c (u · u ) =) c = y·u1 1 1 1 1 1 u1·u1 Similarly, c2 = ,. , cp = Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis: Theorem Theorem (5) Let fu1; u2;:::; upg be an orthogonal basis for a subspace W of Rn: Then each y in W has a unique representation as a linear combination of u1; u2;:::; up. In fact, if y =c1u1 + c2u2 + ··· + cpup then y · uj cj = (j = 1;:::; p) uj · uj Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Basis: Example Example 2 3 3 Express y = 4 7 5 as a linear combination of the orthogonal basis 4 82 3 2 3 2 39 < 1 1 0 = 4 −1 5 ; 4 1 5 ; 4 0 5 . : 0 0 1 ; Solution: y · u y · u y · u 1 = 2 = 3 = u1 · u1 u2 · u2 u3 · u3 Hence y = u1 + u2 + u3 Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Projections For a nonzero vector u in Rn, suppose we want to write y in Rn as the the following y = (multiple of u) + (multiple a vector ? to u) (y − αu) · u =0 =) y · u − α (u · u) =0 =) α = y·u by= u·u u (orthogonal projection of y onto u) y·u z = y− u·u u (component of y orthogonal to u) Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthogonal Projections: Example Example −8 3 Let y = and u = . Compute the distance from y to 4 1 the line through 0 and u. Solution: y · u y= u = b u · u Distance from y to the line through 0 and u = distance from by to y = kby − yk = Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Sets Orthonormal Sets n A set of vectors fu1; u2;:::; upg in R is called an orthonormal set if it is an orthogonal set of unit vectors. Orthonormal Basis If W =spanfu1; u2;:::; upg, then fu1; u2;:::; upg is an orthonormal basis for W : p p Recall that v is a unit vector if kvk = v · v = vT v = 1. Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Matrix: Example Example Suppose U = [u1 u2 u3] where fu1; u2; u3g is an orthonormal set. 2 3 2 3 2 T 3 u1 T T 6 7 6 7 U U = 4 u2 5 [u1 u2 u3] = 6 7 = 6 7 T 4 5 4 5 u3 Orthogonal Matrix It can be shown that UUT = I : So U−1 = UT (such a matrix is called an orthogonal matrix). Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 12 6.2 Orthogonal Sets Orthogonal Sets Basis Projection Orthonormal Matrix Orthonormal Matrix: Theorems Theorem (6) An m × n matrix U has orthonormal columns if and only if UT U = I : Theorem (7) Let U be an m × n matrix with orthonormal columns, and let x and y be in Rn. Then a. kUxk = kxk b.( Ux) · (Uy) = x · y c.( Ux) · (Uy) = 0 if and only if x · y = 0. Proof of part b: (Ux) · (Uy) = Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 12.
Load more

Recommended publications
  • Introduction to Linear Bialgebra
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N.
    [Show full text]
  • 21. Orthonormal Bases
    21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else.
    [Show full text]
  • Orthogonal Bases and the -So in Section 4.8 We Discussed the Problem of Finding the Orthogonal Projection P
    Orthogonal Bases and the -So In Section 4.8 we discussed the problemR. of finding the orthogonal projection p the vector b into the V of , . , suhspace the vectors , v2 ho If v1 v,, form a for V, and the in x n matrix A has these basis vectors as its column vectors. ilt the orthogonal projection p is given by p = Ax where x is the (unique) solution of the normal system ATAx = A7b. The formula for p takes an especially simple and attractive Form when the ba vectors , . .. , v1 v are mutually orthogonal. DEFINITION Orthogonal Basis An orthogonal basis for the suhspacc V of R” is a basis consisting of vectors , ,v,, that are mutually orthogonal, so that v v = 0 if I j. If in additii these basis vectors are unit vectors, so that v1 . = I for i = 1. 2 n, thct the orthogonal basis is called an orthonormal basis. Example 1 The vectors = (1, 1,0), v2 = (1, —1,2), v3 = (—1,1,1) form an orthogonal basis for . We “normalize” ‘ R3 can this orthogonal basis 1w viding each basis vector by its length: If w1=—- (1=1,2,3), lvii 4.9 Orthogonal Bases and the Gram-Schmidt Algorithm 295 then the vectors /1 I 1 /1 1 ‘\ 1 2” / I w1 0) W2 = —— _z) W3 for . form an orthonormal basis R3 , . ..., v, of the m x ii matrix A Now suppose that the column vectors v v2 form an orthogonal basis for the suhspacc V of R’. Then V}.VI 0 0 v2.v ..
    [Show full text]
  • Linear Algebra for Dummies
    Linear Algebra for Dummies Jorge A. Menendez October 6, 2017 Contents 1 Matrices and Vectors1 2 Matrix Multiplication2 3 Matrix Inverse, Pseudo-inverse4 4 Outer products 5 5 Inner Products 5 6 Example: Linear Regression7 7 Eigenstuff 8 8 Example: Covariance Matrices 11 9 Example: PCA 12 10 Useful resources 12 1 Matrices and Vectors An m × n matrix is simply an array of numbers: 2 3 a11 a12 : : : a1n 6 a21 a22 : : : a2n 7 A = 6 7 6 . 7 4 . 5 am1 am2 : : : amn where we define the indexing Aij = aij to designate the component in the ith row and jth column of A. The transpose of a matrix is obtained by flipping the rows with the columns: 2 3 a11 a21 : : : an1 6 a12 a22 : : : an2 7 AT = 6 7 6 . 7 4 . 5 a1m a2m : : : anm T which evidently is now an n × m matrix, with components Aij = Aji = aji. In other words, the transpose is obtained by simply flipping the row and column indeces. One particularly important matrix is called the identity matrix, which is composed of 1’s on the diagonal and 0’s everywhere else: 21 0 ::: 03 60 1 ::: 07 6 7 6. .. .7 4. .5 0 0 ::: 1 1 It is called the identity matrix because the product of any matrix with the identity matrix is identical to itself: AI = A In other words, I is the equivalent of the number 1 for matrices. For our purposes, a vector can simply be thought of as a matrix with one column1: 2 3 a1 6a2 7 a = 6 7 6 .
    [Show full text]
  • Orthonormal Bases in Hilbert Space APPM 5440 Fall 2017 Applied Analysis
    Orthonormal Bases in Hilbert Space APPM 5440 Fall 2017 Applied Analysis Stephen Becker November 18, 2017(v4); v1 Nov 17 2014 and v2 Jan 21 2016 Supplementary notes to our textbook (Hunter and Nachtergaele). These notes generally follow Kreyszig’s book. The reason for these notes is that this is a simpler treatment that is easier to follow; the simplicity is because we generally do not consider uncountable nets, but rather only sequences (which are countable). I have tried to identify the theorems from both books; theorems/lemmas with numbers like 3.3-7 are from Kreyszig. Note: there are two versions of these notes, with and without proofs. The version without proofs will be distributed to the class initially, and the version with proofs will be distributed later (after any potential homework assignments, since some of the proofs maybe assigned as homework). This version does not have proofs. 1 Basic Definitions NOTE: Kreyszig defines inner products to be linear in the first term, and conjugate linear in the second term, so hαx, yi = αhx, yi = hx, αy¯ i. In contrast, Hunter and Nachtergaele define the inner product to be linear in the second term, and conjugate linear in the first term, so hαx,¯ yi = αhx, yi = hx, αyi. We will follow the convention of Hunter and Nachtergaele, so I have re-written the theorems from Kreyszig accordingly whenever the order is important. Of course when working with the real field, order is completely unimportant. Let X be an inner product space. Then we can define a norm on X by kxk = phx, xi, x ∈ X.
    [Show full text]
  • Lecture 4: April 8, 2021 1 Orthogonality and Orthonormality
    Mathematical Toolkit Spring 2021 Lecture 4: April 8, 2021 Lecturer: Avrim Blum (notes based on notes from Madhur Tulsiani) 1 Orthogonality and orthonormality Definition 1.1 Two vectors u, v in an inner product space are said to be orthogonal if hu, vi = 0. A set of vectors S ⊆ V is said to consist of mutually orthogonal vectors if hu, vi = 0 for all u 6= v, u, v 2 S. A set of S ⊆ V is said to be orthonormal if hu, vi = 0 for all u 6= v, u, v 2 S and kuk = 1 for all u 2 S. Proposition 1.2 A set S ⊆ V n f0V g consisting of mutually orthogonal vectors is linearly inde- pendent. Proposition 1.3 (Gram-Schmidt orthogonalization) Given a finite set fv1,..., vng of linearly independent vectors, there exists a set of orthonormal vectors fw1,..., wng such that Span (fw1,..., wng) = Span (fv1,..., vng) . Proof: By induction. The case with one vector is trivial. Given the statement for k vectors and orthonormal fw1,..., wkg such that Span (fw1,..., wkg) = Span (fv1,..., vkg) , define k u + u = v − hw , v i · w and w = k 1 . k+1 k+1 ∑ i k+1 i k+1 k k i=1 uk+1 We can now check that the set fw1,..., wk+1g satisfies the required conditions. Unit length is clear, so let’s check orthogonality: k uk+1, wj = vk+1, wj − ∑ hwi, vk+1i · wi, wj = vk+1, wj − wj, vk+1 = 0. i=1 Corollary 1.4 Every finite dimensional inner product space has an orthonormal basis.
    [Show full text]
  • Ch 5: ORTHOGONALITY
    Ch 5: ORTHOGONALITY 5.5 Orthonormal Sets 1. a set fv1; v2; : : : ; vng is an orthogonal set if hvi; vji = 0, 81 ≤ i 6= j; ≤ n (note that orthogonality only makes sense in an inner product space since you need to inner product to check hvi; vji = 0. n For example fe1; e2; : : : ; eng is an orthogonal set in R . 2. fv1; v2; : : : ; vng is an orthogonal set =) v1; v2; : : : ; vn are linearly independent. 3. a set fv1; v2; : : : ; vng is an orthonormal set if hvi; vji = 0 and jjvijj = 1, 81 ≤ i 6= j; ≤ n (i.e. orthogonal and unit length). n For example fe1; e2; : : : ; eng is an orthonormal set in R . 4. given an orthogonal basis for a vector space V , we can always find an orthonormal basis for V by dividing each vector by its length (see Example 2 and 3 page 256) n 5. a space with an orthonormal basis behaves like the euclidean space R with the standard basis (it is easier to work with basis that are orthonormal, just like it is n easier to work with the standard basis versus other bases for R ) 6. if v is a vector in a inner product space V with fu1; u2; : : : ; ung an orthonormal basis, Pn Pn then we can write v = i=1 ciui = i=1hv; uiiui Pn 7. an easy way to find the inner product of two vectors: if u = i=1 aiui and v = Pn i=1 biui, where fu1; u2; : : : ; ung is an orthonormal basis for an inner product space Pn V , then hu; vi = i=1 aibi Pn 8.
    [Show full text]
  • CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V
    CHAPTER 2 Clifford algebras 1. Exterior algebras 1.1. Definition. For any vector space V over a field K, let T (V ) = k k k Z T (V ) be the tensor algebra, with T (V ) = V V the k-fold tensor∈ product. The quotient of T (V ) by the two-sided⊗···⊗ ideal (V ) generated byL all v w + w v is the exterior algebra, denoted (V ).I The product in (V ) is usually⊗ denoted⊗ α α , although we will frequently∧ omit the wedge ∧ 1 ∧ 2 sign and just write α1α2. Since (V ) is a graded ideal, the exterior algebra inherits a grading I (V )= k(V ) ∧ ∧ k Z M∈ where k(V ) is the image of T k(V ) under the quotient map. Clearly, 0(V )∧ = K and 1(V ) = V so that we can think of V as a subspace of ∧(V ). We may thus∧ think of (V ) as the associative algebra linearly gener- ated∧ by V , subject to the relations∧ vw + wv = 0. We will write φ = k if φ k(V ). The exterior algebra is commutative | | ∈∧ (in the graded sense). That is, for φ k1 (V ) and φ k2 (V ), 1 ∈∧ 2 ∈∧ [φ , φ ] := φ φ + ( 1)k1k2 φ φ = 0. 1 2 1 2 − 2 1 k If V has finite dimension, with basis e1,...,en, the space (V ) has basis ∧ e = e e I i1 · · · ik for all ordered subsets I = i1,...,ik of 1,...,n . (If k = 0, we put { } k { n } e = 1.) In particular, we see that dim (V )= k , and ∅ ∧ n n dim (V )= = 2n.
    [Show full text]
  • Math 22 – Linear Algebra and Its Applications
    Math 22 – Linear Algebra and its applications - Lecture 25 - Instructor: Bjoern Muetzel GENERAL INFORMATION ▪ Office hours: Tu 1-3 pm, Th, Sun 2-4 pm in KH 229 Tutorial: Tu, Th, Sun 7-9 pm in KH 105 ▪ Homework 8: due Wednesday at 4 pm outside KH 008. There is only Section B,C and D. 5 Eigenvalues and Eigenvectors 5.1 EIGENVECTORS AND EIGENVALUES Summary: Given a linear transformation 푇: ℝ푛 → ℝ푛, then there is always a good basis on which the transformation has a very simple form. To find this basis we have to find the eigenvalues of T. GEOMETRIC INTERPRETATION 5 −3 1 1 Example: Let 퐴 = and let 푢 = 푥 = and 푣 = . −6 2 0 2 −1 1.) Find A푣 and Au. Draw a picture of 푣 and A푣 and 푢 and A푢. 2.) Find A(3푢 +2푣) and 퐴2 (3푢 +2푣). Hint: Use part 1.) EIGENVECTORS AND EIGENVALUES ▪ Definition: An eigenvector of an 푛 × 푛 matrix A is a nonzero vector x such that 퐴푥 = 휆푥 for some scalar λ in ℝ. In this case λ is called an eigenvalue and the solution x≠ ퟎ is called an eigenvector corresponding to λ. ▪ Definition: Let A be an 푛 × 푛 matrix. The set of solutions 푛 Eig(A, λ) = {x in ℝ , such that (퐴 − 휆퐼푛)x = 0} is called the eigenspace Eig(A, λ) of A corresponding to λ. It is the null space of the matrix 퐴 − 휆퐼푛: Eig(A, λ) = Nul(퐴 − 휆퐼푛) Slide 5.1- 7 EIGENVECTORS AND EIGENVALUES 16 Example: Show that 휆 =7 is an eigenvalue of matrix A = 52 and find the corresponding eigenspace Eig(A,7).
    [Show full text]
  • Geometric (Clifford) Algebra and Its Applications
    Geometric (Clifford) algebra and its applications Douglas Lundholm F01, KTH January 23, 2006∗ Abstract In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and opera- tions. This view enables us to define Clifford algebras with scalars in arbitrary rings and provides new suggestions for an infinite-dimensional approach. Furthermore, I give a quick review of classic results regarding geo- metric algebras, such as their classification in terms of matrix algebras, the connection to orthogonal and Spin groups, and their representation theory. A number of lower-dimensional examples are worked out in a sys- tematic way using so called norm functions, while general applications of representation theory include normed division algebras and vector fields on spheres. I also consider examples in relativistic physics, where reformulations in terms of geometric algebra give rise to both computational and conceptual simplifications. arXiv:math/0605280v1 [math.RA] 10 May 2006 ∗Corrected May 2, 2006. Contents 1 Introduction 1 2 Foundations 3 2.1 Geometric algebra ( , q)...................... 3 2.2 Combinatorial CliffordG V algebra l(X,R,r)............. 6 2.3 Standardoperations .........................C 9 2.4 Vectorspacegeometry . 13 2.5 Linearfunctions ........................... 16 2.6 Infinite-dimensional Clifford algebra . 19 3 Isomorphisms 23 4 Groups 28 4.1 Group actions on .......................... 28 4.2 TheLipschitzgroupG ......................... 30 4.3 PropertiesofPinandSpingroups . 31 4.4 Spinors ................................ 34 5 A study of lower-dimensional algebras 36 5.1 (R1) ................................. 36 G 5.2 (R0,1) =∼ C -Thecomplexnumbers . 36 5.3 G(R0,0,1)...............................
    [Show full text]
  • True False Questions from 4,5,6
    (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 217: True False Practice Professor Karen Smith 1. For every 2 × 2 matrix A, there exists a 2 × 2 matrix B such that det(A + B) 6= det A + det B. Solution note: False! Take A to be the zero matrix for a counterexample. 2. If the change of basis matrix SA!B = ~e4 ~e3 ~e2 ~e1 , then the elements of A are the same as the element of B, but in a different order. Solution note: True. The matrix tells us that the first element of A is the fourth element of B, the second element of basis A is the third element of B, the third element of basis A is the second element of B, and the the fourth element of basis A is the first element of B. 6 6 3. Every linear transformation R ! R is an isomorphism. Solution note: False. The zero map is a a counter example. 4. If matrix B is obtained from A by swapping two rows and det A < det B, then A is invertible. Solution note: True: we only need to know det A 6= 0. But if det A = 0, then det B = − det A = 0, so det A < det B does not hold. 5. If two n × n matrices have the same determinant, they are similar. Solution note: False: The only matrix similar to the zero matrix is itself. But many 1 0 matrices besides the zero matrix have determinant zero, such as : 0 0 6.
    [Show full text]
  • C Self-Adjoint Operators and Complete Orthonormal Bases
    C Self-adjoint operators and complete orthonormal bases The concept of the adjoint of an operator plays a very important role in many aspects of linear algebra and functional analysis. Here we will primar­ ily focus on s e ll~adjoint operators and how they can be used to obtain and characterize complete orthonormal sets of vectors in a Hilbert space. The main fact we will present here is that self-adjoint operators typically have orthogonal eigenvectors, and under some conditions, these eigenvectors can form a complete orthonormal basis for the Hilbert space. In particular, we will use this technique to find a variety of orthonormal bases for L2 [a, b] that satisfy different types of boundary conditions. We begin with the algebraic aspects of self-adjoint operators and their eigenvectors. Those algebraic properties are identical in the finite and infinite dimensional cases. The completeness arguments in infinite dimensions are more delicate, we will only state those results without complete proofs. We begin first by defining the adjoint. D efinition C.l. (Finite dimensional or bounded operator case) Let A H I -; H2 be a bounded operator between the Hilbert spaces HI and H 2. The adjomt A' : H2 ---; HI of Xis defined by the req1Lirement J (V2, AVI ) ~ = (A''V2, v])], (C.l) fOT all VI E HI--- and'V2 E H2· Note that for emphasis, we have written (. , .)] and (. , ')2 for the inner products in HI and H2 respectively. It remains to be shown that the relation (C.l) defines an operator A' from A uniquely. We will not do this here in general but will illustrate it with several examples.
    [Show full text]