DOCSLIB.ORG

Lecture 8 Applications of Vectors and Matrces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 8 Applications of Vectors and Matrces Lecture 8: Vectors and Matrices III – Applications of Vectors and Matrices (See Sections 3.2, 3.3 and 3.8 to 3.12 in Boas) As noted in the previous lecture the notation of vectors and matrices is useful not only for the study of linear transformations but also for (the related) problem of solving systems of simultaneous linear equations. Consider N simultaneous linear equations for N unknowns, xk , k 1, , N , A111 x A 122 x A 111NNNN x A 1 x X 1 . (8.1) AN1 x 1 A N 2 x 2 A NN 1 x N 1 A NN x N X N Clearly we are encouraged to think in terms of a NxN matrix representation for the coefficients, Akl , a N-D vector of unknowns, xk , and a N-D vector of known inputs, X l . Then, using the notation we have developed (to simplify our lives), we can write Ax X. (8.2) As long as the matrix A is nonsingular, detA 0 , we can find its inverse and solve the simultaneous equations in matrix form ( CA is the cofactor matrix for A) as defined in Eq. 7.29 in the previous lecture, T NNNCA 11 klXC l lk x A X xk A kl X l X l l1 l 1det AA l 1 det det A (8.3) x k . k det A In the last expression the matrix A is the matrix A with the kth column replaced by k the column vector X , AAXAA11 1k 1 1 1 k 1 1 N A . k (8.4) AAXAAN1 Nk 1 N Nk 1 NN Physics 227 Lecture 8 1 Autumn 2008 We recognize that Eqs. (8.3) and (8.4) define Cramer’s rule (see Section 3.3 in Boas). Next let us consider the more general problem where Eq. (8.2) corresponds to M equations for N unknowns. The vector now has M components, the vector x has N components, and the matrix A is MxN (M rows and N columns). The question to ask first is - when does this system of equations have a solution? This corresponds to determining how many linearly independent equations are present in this revised version of Eq. (8.2). To answer this question we perform the process known as row reduction (i.e., add/subtract rows times constants to/from other rows and interchange rows, See Boas 3.2) to find a new version of the matrix A with as many zero elements as possible in the lower left hand corner (“below the pivot”). This version of A is called the row echelon form. Having performed this process (to the maximum extent possible) we count the number of rows in the reduced matrix (still a MxN matrix) with nonzero elements. Call this number M and label it the rank of the matrix A, where clearly MM . The rank counts the number of linearly independent rows in A and thus the number of linearly independent equations in the original system of equations. ASIDE: We can apply this same idea of linear independence to a general set of functions (not just linear functions). Consider the determinant of the matrix defined in terms of N functions, fk x, k 1, , N , and their derivatives (with respect to x) f12 x f x fN x f x f x f x Wx 12 N . (8.5) NNN1 1 1 f12 f fN This determinant is called the Wronskian of the functions. At values of x where Wx 0, the functions are linearly independent in the sense that (recall our discussion of linearly independent vectors) X N ck f k x 0 c k 0, k 1, , N . (8.6) k1 No linear combination of the functions vanishes except the trivial one (all zero coefficients) in the range of x where . Physics 227 Lecture 8 2 Autumn 2008 Returning to the problem of simultaneous equations we define a new Mx(N+1) matrix, the augmented version of A, which has the column vector as its N 1th column, AAX11 1N 1 A . Aug (8.7) AAXM1 MN M Next we row reduce this matrix to determine its rank. Call this last quantity M Aug and note that cannot be smaller than M , MMMAug , since every row that can be reduced to zeros in AAug can certainly be reduced to zeros in A. Now we can list the various possibilities. 1) If MM Aug (the rank of A is smaller than the rank of ), the equations are inconsistent with each other and there are no solutions to the set of simultaneous equations. 2) If MMNMAug , we can eliminate MM non-independent equations (rows) from the system of equations and arrive at the NxN problem we discussed at the beginning of the lecture. The resulting NxN matrix of coefficients we label ARed , the “reduced” matrix. Since we have eliminated all the rows that have only zero elements, we are guaranteed that the determinant -1 of the reduced matrix, , is nonzero and that an inverse matrix, ARed , exists. Hence we can use this inverse, i.e., Cramer’s rule, to find a unique solution to the system of equations, i.e., a unique value for the N-D column vector -1 x ARed X . 3) If MMNAug , the reduced problem has fewer equations than there are unknowns, i.e., there are not enough independent equations to determine all of the unknowns. Rather the solution to this system corresponds to solving for X M of the unknowns in terms of the other NM unknowns (and the original coefficients and the known vector X ). As an example consider the equations Physics 227 Lecture 8 3 Autumn 2008 x y 2 z 7 7 1 1 2 2x y 4 z 2 M 4 2 2 1 4 ,,.XA 5410x y z 1 N 3 1 5410 (8.8) 3x y 6 z 5 5 3 1 6 Proceeding to row reduce we have 1 1 2 1 1 2 0 3 0 0 3 0 A . RRRR2 2 10 9 0 3 3 2 0 0 0 (8.9) RRRR3 5 1 4 4 3 2 RR4 3 1 0 4 0 0 0 0 Hence we see that the rank of A is M 2 . Proceeding with the augmented matrix we have 1 1 2 7 1 1 2 7 2 1 4 2 0 3 0 12 A Aug 5 4 10 1 RR2 2 1 0 9 0 36 RR3 5 1 3 1 6 5 RR4 3 1 0 4 0 16 1 1 2 7 (8.10) 0 3 0 12 . RR3 3 2 0 0 0 0 RR4 4 3 2 0 0 0 0 Thus we have MMNAug 23 and the problem is under-constrained. We can solve for 2 unknowns in terms of the third. From the revised equations we have x y 2 z 7 x 7 y 2 z 3 2 z . (8.11) 3yy 12 4 The unknown z remains a free variable. The choice of z instead of x to play this role is, of course, arbitrary. Physics 227 Lecture 8 4 Autumn 2008 Now return to Eq. (8.2) and think about it in a slightly different way. What if the right-side-side ( X ) is proportional to the unknown vector x , i.e., can we find a vector such that after we perform the linear transform represented by the matrix A we end up back at times a constant? As we will see, the problem of finding vectors that are invariant (except for a overall constant) under a given transformation is extremely important/useful in physics (here we move to Section 3.11 in Boas, or Chapter 10 in the previous edition of Boas). We want to consider the following new version of Eq. (8.2), Ax x A 1 x 0. (8.12) Note that this version is a homogenous equation. If the matrix A 1 is 1 nonsingular and A 1 exists, we know that the above equation, Eq. (8.12), has only the trivial solution 1 xA 1 0 0. (8.13) Thus, to obtain interesting solutions (and we want to do that), we require that the inverse not exist or that detA1 0. (8.14) This condition, the eigenvalue condition (also called the characteristic equation for A), is an Nth order equation for the parameter generally having N distinct solutions, which may be complex. Note that the eigenvalues are a property of the transformation represented by A, and not of the vectors being transformed. On the other hand, for each specific eigenvalue there is a corresponding vector, the k eigenvectorvk , which satisfies Avk k v k . (8.15) Since this is a homogenous equation, the length of v is not specified (if is a k solution so is cvk , where c is a constant) and typically we choose to make it a unit T vector (to simplify the analysis – the lazy but smart argument), vk v k v k v k 1. (Another way to see this point is to note from Eq. (8.14) that only N-1 of the N Physics 227 Lecture 8 5 Autumn 2008 equations in (8.15) are linearly independent. You should see this explicitly as you solve (8.15).) For complex vectors the normalization choice is typically T † vk v k v k v k v k v k 1, which leaves an overall phase undetermined (i.e., i this constraint is invariant under vkk e v ). We can derive a few important properties of eigenvalues and vectors by considering a pair of distinct eigenvalues, kl, , and the corresponding eigenvectors, vvkl, for a general complex matrix A.
Load more

Recommended publications
  • Math 4571 (Advanced Linear Algebra) Lecture #27
    Math 4571 (Advanced Linear Algebra) Lecture #27 Applications of Diagonalization and the Jordan Canonical Form (Part 1): Spectral Mapping and the Cayley-Hamilton Theorem Transition Matrices and Markov Chains The Spectral Theorem for Hermitian Operators This material represents x4.4.1 + x4.4.4 +x4.4.5 from the course notes. Overview In this lecture and the next, we discuss a variety of applications of diagonalization and the Jordan canonical form. This lecture will discuss three essentially unrelated topics: A proof of the Cayley-Hamilton theorem for general matrices Transition matrices and Markov chains, used for modeling iterated changes in systems over time The spectral theorem for Hermitian operators, in which we establish that Hermitian operators (i.e., operators with T ∗ = T ) are diagonalizable In the next lecture, we will discuss another fundamental application: solving systems of linear differential equations. Cayley-Hamilton, I First, we establish the Cayley-Hamilton theorem for arbitrary matrices: Theorem (Cayley-Hamilton) If p(x) is the characteristic polynomial of a matrix A, then p(A) is the zero matrix 0. The same result holds for the characteristic polynomial of a linear operator T : V ! V on a finite-dimensional vector space. Cayley-Hamilton, II Proof: Since the characteristic polynomial of a matrix does not depend on the underlying field of coefficients, we may assume that the characteristic polynomial factors completely over the field (i.e., that all of the eigenvalues of A lie in the field) by replacing the field with its algebraic closure. Then by our results, the Jordan canonical form of A exists.
    [Show full text]
  • Quantum Information
    Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding .
    [Show full text]
  • Lecture Notes: Qubit Representations and Rotations
    Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1.
    [Show full text]
  • MATH 2370, Practice Problems
    MATH 2370, Practice Problems Kiumars Kaveh Problem: Prove that an n × n complex matrix A is diagonalizable if and only if there is a basis consisting of eigenvectors of A. Problem: Let A : V ! W be a one-to-one linear map between two finite dimensional vector spaces V and W . Show that the dual map A0 : W 0 ! V 0 is surjective. Problem: Determine if the curve 2 2 2 f(x; y) 2 R j x + y + xy = 10g is an ellipse or hyperbola or union of two lines. Problem: Show that if a nilpotent matrix is diagonalizable then it is the zero matrix. Problem: Let P be a permutation matrix. Show that P is diagonalizable. Show that if λ is an eigenvalue of P then for some integer m > 0 we have λm = 1 (i.e. λ is an m-th root of unity). Hint: Note that P m = I for some integer m > 0. Problem: Show that if λ is an eigenvector of an orthogonal matrix A then jλj = 1. n Problem: Take a vector v 2 R and let H be the hyperplane orthogonal n n to v. Let R : R ! R be the reflection with respect to a hyperplane H. Prove that R is a diagonalizable linear map. Problem: Prove that if λ1; λ2 are distinct eigenvalues of a complex matrix A then the intersection of the generalized eigenspaces Eλ1 and Eλ2 is zero (this is part of the Spectral Theorem). 1 Problem: Let H = (hij) be a 2 × 2 Hermitian matrix. Use the Min- imax Principle to show that if λ1 ≤ λ2 are the eigenvalues of H then λ1 ≤ h11 ≤ λ2.
    [Show full text]
  • A Representations of SU(2)
    A Representations of SU(2) In this appendix we provide details of the parameterization of the group SU(2) and differential forms on the group space. An arbitrary representation of the group SU(2) is given by the set of three generators Tk, which satisfy the Lie algebra [Ti,Tj ]=iεijkTk, with ε123 =1. The element of the group is given by the matrix U =exp{iT · ω} , (A.1) 1 where in the fundamental representation Tk = 2 σk, k =1, 2, 3, with 01 0 −i 10 σ = ,σ= ,σ= , 1 10 2 i 0 3 0 −1 standard Pauli matrices, which satisfy the relation σiσj = δij + iεijkσk . The vector ω has components ωk in a given coordinate frame. Geometrically, the matrices U are generators of spinor rotations in three- 3 dimensional space R and the parameters ωk are the corresponding angles of rotation. The Euler parameterization of an arbitrary matrix of SU(2) transformation is defined in terms of three angles θ, ϕ and ψ,as ϕ θ ψ iσ3 iσ2 iσ3 U(ϕ, θ, ψ)=Uz(ϕ)Uy(θ)Uz(ψ)=e 2 e 2 e 2 ϕ ψ ei 2 0 cos θ sin θ ei 2 0 = 2 2 −i ϕ θ θ − ψ 0 e 2 − sin cos 0 e i 2 2 2 i i θ 2 (ψ+ϕ) θ − 2 (ψ−ϕ) cos 2 e sin 2 e = i − − i . (A.2) − θ 2 (ψ ϕ) θ 2 (ψ+ϕ) sin 2 e cos 2 e Thus, the SU(2) group manifold is isomorphic to three-sphere S3.
    [Show full text]
  • Multipartite Quantum States and Their Marginals
    Diss. ETH No. 22051 Multipartite Quantum States and their Marginals A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by Michael Walter Diplom-Mathematiker arXiv:1410.6820v1 [quant-ph] 24 Oct 2014 Georg-August-Universität Göttingen born May 3, 1985 citizen of Germany accepted on the recommendation of Prof. Dr. Matthias Christandl, examiner Prof. Dr. Gian Michele Graf, co-examiner Prof. Dr. Aram Harrow, co-examiner 2014 Abstract Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions – such as Pauli’s exclusion principle or the monogamy of quantum entanglement – which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part of this thesis (Chapters 2–6) we focus on the one-body marginals of multipartite quantum states. Starting from a novel ge- ometric solution of the compatibility problem, we then turn towards the phenomenon of quantum entanglement. We find that the one-body marginals through their local eigenvalues can characterize the entan- glement of multipartite quantum states, and we propose the notion of an entanglement polytope for its systematic study.
    [Show full text]
  • Theory of Angular Momentum and Spin
    Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO(3) of the associated rotation matrices and the 1 corresponding transformation matrices of spin{ 2 states forming the group SU(2) occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors ~r in 3-dimensional space, i.e., ~r R3, 2 we will then consider transformations of wavefunctions (~r) of single particles in R3, and finally N transformations of products of wavefunctions like j(~rj) which represent a system of N (spin- Qj=1 zero) particles in R3. We will also review below the well-known fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states.
    [Show full text]
  • Two-State Systems
    1 TWO-STATE SYSTEMS Introduction. Relative to some/any discretely indexed orthonormal basis |n) | ∂ | the abstract Schr¨odinger equation H ψ)=i ∂t ψ) can be represented | | | ∂ | (m H n)(n ψ)=i ∂t(m ψ) n ∂ which can be notated Hmnψn = i ∂tψm n H | ∂ | or again ψ = i ∂t ψ We found it to be the fundamental commutation relation [x, p]=i I which forced the matrices/vectors thus encountered to be ∞-dimensional. If we are willing • to live without continuous spectra (therefore without x) • to live without analogs/implications of the fundamental commutator then it becomes possible to contemplate “toy quantum theories” in which all matrices/vectors are finite-dimensional. One loses some physics, it need hardly be said, but surprisingly much of genuine physical interest does survive. And one gains the advantage of sharpened analytical power: “finite-dimensional quantum mechanics” provides a methodological laboratory in which, not infrequently, the essentials of complicated computational procedures can be exposed with closed-form transparency. Finally, the toy theory serves to identify some unanticipated formal links—permitting ideas to flow back and forth— between quantum mechanics and other branches of physics. Here we will carry the technique to the limit: we will look to “2-dimensional quantum mechanics.” The theory preserves the linearity that dominates the full-blown theory, and is of the least-possible size in which it is possible for the effects of non-commutivity to become manifest. 2 Quantum theory of 2-state systems We have seen that quantum mechanics can be portrayed as a theory in which • states are represented by self-adjoint linear operators ρ ; • motion is generated by self-adjoint linear operators H; • measurement devices are represented by self-adjoint linear operators A.
    [Show full text]
  • 10 Group Theory
    10 Group theory 10.1 What is a group? A group G is a set of elements f, g, h, ... and an operation called multipli- cation such that for all elements f,g, and h in the group G: 1. The product fg is in the group G (closure); 2. f(gh)=(fg)h (associativity); 3. there is an identity element e in the group G such that ge = eg = g; 1 1 1 4. every g in G has an inverse g− in G such that gg− = g− g = e. Physical transformations naturally form groups. The elements of a group might be all physical transformations on a given set of objects that leave invariant a chosen property of the set of objects. For instance, the objects might be the points (x, y) in a plane. The chosen property could be their distances x2 + y2 from the origin. The physical transformations that leave unchanged these distances are the rotations about the origin p x cos ✓ sin ✓ x 0 = . (10.1) y sin ✓ cos ✓ y ✓ 0◆ ✓− ◆✓ ◆ These rotations form the special orthogonal group in 2 dimensions, SO(2). More generally, suppose the transformations T,T0,T00,... change a set of objects in ways that leave invariant a chosen property property of the objects. Suppose the product T 0 T of the transformations T and T 0 represents the action of T followed by the action of T 0 on the objects. Since both T and T 0 leave the chosen property unchanged, so will their product T 0 T . Thus the closure condition is satisfied.
    [Show full text]
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation
    [Show full text]
  • Unipotent Flows and Applications
    Clay Mathematics Proceedings Volume 10, 2010 Unipotent Flows and Applications Alex Eskin 1. General introduction 1.1. Values of indefinite quadratic forms at integral points. The Op- penheim Conjecture. Let X Q(x1; : : : ; xn) = aijxixj 1≤i≤j≤n be a quadratic form in n variables. We always assume that Q is indefinite so that (so that there exists p with 1 ≤ p < n so that after a linear change of variables, Q can be expresses as: Xp Xn ∗ 2 − 2 Qp(y1; : : : ; yn) = yi yi i=1 i=p+1 We should think of the coefficients aij of Q as real numbers (not necessarily rational or integer). One can still ask what will happen if one substitutes integers for the xi. It is easy to see that if Q is a multiple of a form with rational coefficients, then the set of values Q(Zn) is a discrete subset of R. Much deeper is the following conjecture: Conjecture 1.1 (Oppenheim, 1929). Suppose Q is not proportional to a ra- tional form and n ≥ 5. Then Q(Zn) is dense in the real line. This conjecture was extended by Davenport to n ≥ 3. Theorem 1.2 (Margulis, 1986). The Oppenheim Conjecture is true as long as n ≥ 3. Thus, if n ≥ 3 and Q is not proportional to a rational form, then Q(Zn) is dense in R. This theorem is a triumph of ergodic theory. Before Margulis, the Oppenheim Conjecture was attacked by analytic number theory methods. (In particular it was known for n ≥ 21, and for diagonal forms with n ≥ 5).
    [Show full text]
  • Math 223 Symmetric and Hermitian Matrices. Richard Anstee an N × N Matrix Q Is Orthogonal If QT = Q−1
    Math 223 Symmetric and Hermitian Matrices. Richard Anstee An n × n matrix Q is orthogonal if QT = Q−1. The columns of Q would form an orthonormal basis for Rn. The rows would also form an orthonormal basis for Rn. A matrix A is symmetric if AT = A. Theorem 0.1 Let A be a symmetric n × n matrix of real entries. Then there is an orthogonal matrix Q and a diagonal matrix D so that AQ = QD; i.e. QT AQ = D: Note that the entries of M and D are real. There are various consequences to this result: A symmetric matrix A is diagonalizable A symmetric matrix A has an othonormal basis of eigenvectors. A symmetric matrix A has real eigenvalues. Proof: The proof begins with an appeal to the fundamental theorem of algebra applied to det(A − λI) which asserts that the polynomial factors into linear factors and one of which yields an eigenvalue λ which may not be real. Our second step it to show λ is real. Let x be an eigenvector for λ so that Ax = λx. Again, if λ is not real we must allow for the possibility that x is not a real vector. Let xH = xT denote the conjugate transpose. It applies to matrices as AH = AT . Now xH x ≥ 0 with xH x = 0 if and only if x = 0. We compute xH Ax = xH (λx) = λxH x. Now taking complex conjugates and transpose (xH Ax)H = xH AH x using that (xH )H = x. Then (xH Ax)H = xH Ax = λxH x using AH = A.
    [Show full text]