DOCSLIB.ORG

Orthogonal Matrices, Change of Basis, Rank

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Orthogonal Matrices, Change of Basis, Rank Orthogonal matrices, change of basis, rank Math Tools for Neuroscience (NEU 314) Spring 2016 Jonathan Pillow Princeton Neuroscience Institute & Psychology. Lecture 6 (Thursday 2/18) accompanying notes/slides today’s topics • orthonormal basis • change of basis • orthogonal matrix • rank • column space and row space • null space is clearly a vector space [verify]. Working backwards, a set of vectors is said to span a vector space if one can write any v vector in the vector space as a linear com- 1 v3 bination of the set. A spanning set can be redundant: For example, if two of the vec- tors are identical, or are scaled copies of each other. This redundancy is formalized by defining linear independence.Asetofvec- tors {⃗v1,⃗v2,...⃗vM } is linearly independent if v2 (and only if) the only solution to the equation αn⃗vn =0 !n is αn =0(for all n). basis A basis for a vector space is a linearly in- dependent spanning set. For example, con- • set of vectors that can “span” (form via linear combination) all points in a vector space sider the plane of this page. One vector is not enough to span the plane. Scalar multi- v ples of this vector will trace out a line (which v v v 2 is a subspace), but cannot “get off the line” vv 2 1 v v to cover the rest of the plane. But two vec- 1 v v tors are sufficient to span the entire plane. 1 Bases are not unique: any two vectors will do, as long as they don’t lie along the same 1D vector space Two different (orthonormal) line. Three vectors are redundant: one can (subspace of R2) bases for the same 2D always be written as a linear combination of vector space the other two. In general, the vector space N R requires a basis of size N. e Geometrically, the basis vectors define a set of coordinate axes for the space (although e ˆ2 they need not be perpendicular). The stan- e dard basis is the set of unit vectors that lie along the axes of the space: 1 0 0 eˆ1 ⎛ 0 ⎞ ⎛ 1 ⎞ ⎛ 0 ⎞ eˆ1 = ⎜ 0 ⎟, eˆ2 = ⎜ 0 ⎟,...eˆN = ⎜ 0 ⎟. ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ v ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ S x ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 5 x S e ) v x e x S x S e x e v x S e S x) v x e x S x S e x e o o o o o 3 o o o o o 3 orthonormal basis • basis composed of orthogonal unit vectors Change of basis • Let B denote a matrix whose columns form an orthonormal basis for a vector space W If B is full rank (n x n), then we can get back to the original basis through multiplication by B Change of basis • Let B denote a matrix whose columns form an orthonormal basis for a vector space W Vector of projections of v along each basis vector Orthogonal matrix • In this case (full rank, orthogonal columns), B is an orthogonal matrix Properties: length- preserving nothing. This matrix is called the identity,denotedI. If an element of the diagonal is zero, then the associated axis is annihilated. The set of vectors that are annihilated by the matrix form a vector space [prove], which is called the row nullspace,orsimplythenullspace of the matrix. v v 2 0 2 [ 0 0 ] v1 2v1 Another implication of a zero diagonal element is that the matrix cannot “reach” the entire output space, but only a proper subspace. This space is called the column space of the matrix, since it is spanned by the matrix columns. The rank of a matrix is just the dimensionality of the column space. A matrix is said to have full rankOrthogonalif its rank is equal to matrix the smaller of its two dimensions. An orthogonal matrix is a square• matrix2D example: rotation matrix whose columns are pairwise orthogonal unit vectors. Remember that the columns of a matrix describe the response of the system to e^ the standard basis. Thus an orthogonal ma- 2 ( Ο ) = trix maps the standard basis onto a new set e^ 1 ^ of N orthogonal axes, which form an alter- ^ Ο(e1 ) Ο (e2 ) native basis for the space. This operation is cos θ sin θ a generalized rotation, since it corresponds to e g .. Ο = sin θ cosθ] a physical rotation of the space and possibly [ negation of some axes. Thus, the product of two orthogonal matrices is also orthogonal. Note that an orthogonal is full rank (it has no nullspace), since a rotation cannot annihilate any non-zero vector. Linear Systems of Equations The classic motivation for the study of linear algebra is the solution of sets of linear equations such as a11v1 + a12v2 + ...+ a1N vN = b1 a21v1 + a22v2 + ...+ a2N vN = b2 . aM1v1 + aM2v2 + ...+ aMNvN = bM 8 Rank • the rank of a matrix is equal to • # of linearly independent columns • # of linearly independent rows (remarkably, these are always the same) equivalent definition: • the rank of a matrix is the dimensionality of the vector space spanned by its rows or its columns for an m x n matrix A: rank(A) ≤ min(m,n) (can’t be greater than # of rows or # of columns) column space of a matrix W: n × m matrix vector space spanned by the columns of W c1 … cm • these vectors live in an n-dimensional space, so the column space is a subspace of Rn row space of a matrix W: n × m matrix r1 vector space spanned by the … rows of W rn • these vectors live in an m-dimensional space, so the column space is a subspace of Rm null space of a matrix W: n × m matrix • the vector space consisting of r1 all vectors that are orthogonal to … the rows of W rn • equivalently: the null space of W is the vector space of all vectors x such that Wx = 0. • the null space is therefore entirely orthogonal to the row space of a matrix. Together, they make up all of Rm. is clearly a vector space [verify]. Working backwards, a set of vectors is said to span a vector space if one can write any v vector in the vector space as a linear com- 1 v3 bination of the set. A spanning set can be redundant: For example, if two of the vec- tors are identical, or are scaled copies of each other. This redundancy is formalized by defining linear independence.Asetofvec- tors {⃗v1,⃗v2,...⃗vM } is linearly independent if v2 (and only if) the only solution to the equation αn⃗vn =0 n ! null space of a matrix W: is αn =0(for all n). W = ( v 1 ) A basis for a vector space is a linearly in- dependent spanning set. For example, con-null space sider the plane of this page. One vector is 1D vector space not enough to span the plane. Scalar multi- spanned by v1 v ples of this vector will trace out a line (which v v v 2 is a subspace), but cannot “get off the line” vv 2 basis for null space 1 v v to cover the rest of the plane. But two vec- 1 v v tors are sufficient to span the entire plane. 1 Bases are not unique: any two vectors will do, as long as they don’t lie along the same line. Three vectors are redundant: one can always be written as a linear combination of the other two. In general, the vector space N R requires a basis of size N. e Geometrically, the basis vectors define a set of coordinate axes for the space (although e ˆ2 they need not be perpendicular). The stan- e dard basis is the set of unit vectors that lie along the axes of the space: 1 0 0 eˆ1 ⎛ 0 ⎞ ⎛ 1 ⎞ ⎛ 0 ⎞ eˆ1 = ⎜ 0 ⎟, eˆ2 = ⎜ 0 ⎟,...eˆN = ⎜ 0 ⎟. ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ v ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ S x ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 5 x S e ) v x e x S x S e x e v x S e S x) v x e x S x S e x e o o o o o 3 o o o o o 3.
Load more

Recommended publications
  • 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 Reduction 1 the Row Echelon Form -.: Mathematical
    MATH 5330: Computational Methods of Linear Algebra Lecture 9: Orthogonal Reduction Xianyi Zeng Department of Mathematical Sciences, UTEP 1 The Row Echelon Form Our target is to solve the normal equation: AtAx = Atb ; (1.1) m×n where A 2 R is arbitrary; we have shown previously that this is equivalent to the least squares problem: min jjAx−bjj : (1.2) x2Rn t n×n As A A2R is symmetric positive semi-definite, we can try to compute the Cholesky decom- t t n×n position such that A A = L L for some lower-triangular matrix L 2 R . One problem with this approach is that we're not fully exploring our information, particularly in Cholesky decomposition we treat AtA as a single entity in ignorance of the information about A itself. t m×m In particular, the structure A A motivates us to study a factorization A=QE, where Q2R m×n is orthogonal and E 2 R is to be determined. Then we may transform the normal equation to: EtEx = EtQtb ; (1.3) t m×m where the identity Q Q = Im (the identity matrix in R ) is used. This normal equation is equivalent to the least squares problem with E: t min Ex−Q b : (1.4) x2Rn Because orthogonal transformation preserves the L2-norm, (1.2) and (1.4) are equivalent to each n other. Indeed, for any x 2 R : jjAx−bjj2 = (b−Ax)t(b−Ax) = (b−QEx)t(b−QEx) = [Q(Qtb−Ex)]t[Q(Qtb−Ex)] t t t t t t t t 2 = (Q b−Ex) Q Q(Q b−Ex) = (Q b−Ex) (Q b−Ex) = Ex−Q b : Hence the target is to find an E such that (1.3) is easier to solve.
    [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]
  • On the Eigenvalues of Euclidean Distance Matrices
    “main” — 2008/10/13 — 23:12 — page 237 — #1 Volume 27, N. 3, pp. 237–250, 2008 Copyright © 2008 SBMAC ISSN 0101-8205 www.scielo.br/cam On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH∗ Department of Mathematics and Statistics University of Windsor, Windsor, Ontario N9B 3P4, Canada E-mail: [email protected] Abstract. In this paper, the notion of equitable partitions (EP) is used to study the eigenvalues of Euclidean distance matrices (EDMs). In particular, EP is used to obtain the characteristic poly- nomials of regular EDMs and non-spherical centrally symmetric EDMs. The paper also presents methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues. Mathematical subject classification: 51K05, 15A18, 05C50. Key words: Euclidean distance matrices, eigenvalues, equitable partitions, characteristic poly- nomial. 1 Introduction ( ) An n ×n nonzero matrix D = di j is called a Euclidean distance matrix (EDM) 1, 2,..., n r if there exist points p p p in some Euclidean space < such that i j 2 , ,..., , di j = ||p − p || for all i j = 1 n where || || denotes the Euclidean norm. i , ,..., Let p , i ∈ N = {1 2 n}, be the set of points that generate an EDM π π ( , ,..., ) D. An m-partition of D is an ordered sequence = N1 N2 Nm of ,..., nonempty disjoint subsets of N whose union is N. The subsets N1 Nm are called the cells of the partition. The n-partition of D where each cell consists #760/08. Received: 07/IV/08. Accepted: 17/VI/08. ∗Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS.
    [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]
  • 4.1 RANK of a MATRIX Rank List Given Matrix M, the Following Are Equal
    page 1 of Section 4.1 CHAPTER 4 MATRICES CONTINUED SECTION 4.1 RANK OF A MATRIX rank list Given matrix M, the following are equal: (1) maximal number of ind cols (i.e., dim of the col space of M) (2) maximal number of ind rows (i.e., dim of the row space of M) (3) number of cols with pivots in the echelon form of M (4) number of nonzero rows in the echelon form of M You know that (1) = (3) and (2) = (4) from Section 3.1. To see that (3) = (4) just stare at some echelon forms. This one number (that all four things equal) is called the rank of M. As a special case, a zero matrix is said to have rank 0. how row ops affect rank Row ops don't change the rank because they don't change the max number of ind cols or rows. example 1 12-10 24-20 has rank 1 (maximally one ind col, by inspection) 48-40 000 []000 has rank 0 example 2 2540 0001 LetM= -2 1 -1 0 21170 To find the rank of M, use row ops R3=R1+R3 R4=-R1+R4 R2ØR3 R4=-R2+R4 2540 0630 to get the unreduced echelon form 0001 0000 Cols 1,2,4 have pivots. So the rank of M is 3. how the rank is limited by the size of the matrix IfAis7≈4then its rank is either 0 (if it's the zero matrix), 1, 2, 3 or 4. The rank can't be 5 or larger because there can't be 5 ind cols when there are only 4 cols to begin with.
    [Show 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.
    [Show full text]
  • Coordinatization
    MATH 355 Supplemental Notes Coordinatization Coordinatization In R3, we have the standard basis i, j and k. When we write a vector in coordinate form, say 3 v 2 , (1) “ »´ fi 5 — ffi – fl it is understood as v 3i 2j 5k. “ ´ ` The numbers 3, 2 and 5 are the coordinates of v relative to the standard basis ⇠ i, j, k . It has p´ q “p q always been understood that a coordinate representation such as that in (1) is with respect to the ordered basis ⇠. A little thought reveals that it need not be so. One could have chosen the same basis elements in a di↵erent order, as in the basis ⇠ i, k, j . We employ notation indicating the 1 “p q coordinates are with respect to the di↵erent basis ⇠1: 3 v 5 , to mean that v 3i 5k 2j, r s⇠1 “ » fi “ ` ´ 2 —´ ffi – fl reflecting the order in which the basis elements fall in ⇠1. Of course, one could employ similar notation even when the coordinates are expressed in terms of the standard basis, writing v for r s⇠ (1), but whenever we have coordinatization with respect to the standard basis of Rn in mind, we will consider the wrapper to be optional. r¨s⇠ Of course, there are many non-standard bases of Rn. In fact, any linearly independent collection of n vectors in Rn provides a basis. Say we take 1 1 1 4 ´ ´ » 0fi » 1fi » 1fi » 1fi u , u u u ´ . 1 “ 2 “ 3 “ 4 “ — 3ffi — 1ffi — 0ffi — 2ffi — ffi —´ ffi — ffi — ffi — 0ffi — 4ffi — 2ffi — 1ffi — ffi — ffi — ffi —´ ffi – fl – fl – fl – fl As per the discussion above, these vectors are being expressed relative to the standard basis of R4.
    [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]
  • §9.2 Orthogonal Matrices and Similarity Transformations
    n×n Thm: Suppose matrix Q 2 R is orthogonal. Then −1 T I Q is invertible with Q = Q . n T T I For any x; y 2 R ,(Q x) (Q y) = x y. n I For any x 2 R , kQ xk2 = kxk2. Ex 0 1 1 1 1 1 2 2 2 2 B C B 1 1 1 1 C B − 2 2 − 2 2 C B C T H = B C ; H H = I : B C B − 1 − 1 1 1 C B 2 2 2 2 C @ A 1 1 1 1 2 − 2 − 2 2 x9.2 Orthogonal Matrices and Similarity Transformations n×n Def: A matrix Q 2 R is said to be orthogonal if its columns n (1) (2) (n)o n q ; q ; ··· ; q form an orthonormal set in R . Ex 0 1 1 1 1 1 2 2 2 2 B C B 1 1 1 1 C B − 2 2 − 2 2 C B C T H = B C ; H H = I : B C B − 1 − 1 1 1 C B 2 2 2 2 C @ A 1 1 1 1 2 − 2 − 2 2 x9.2 Orthogonal Matrices and Similarity Transformations n×n Def: A matrix Q 2 R is said to be orthogonal if its columns n (1) (2) (n)o n q ; q ; ··· ; q form an orthonormal set in R . n×n Thm: Suppose matrix Q 2 R is orthogonal. Then −1 T I Q is invertible with Q = Q . n T T I For any x; y 2 R ,(Q x) (Q y) = x y.
    [Show full text]
  • The Jordan Canonical Forms of Complex Orthogonal and Skew-Symmetric Matrices Roger A
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Linear Algebra and its Applications 302–303 (1999) 411–421 www.elsevier.com/locate/laa The Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices Roger A. Horn a,∗, Dennis I. Merino b aDepartment of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112–0090, USA bDepartment of Mathematics, Southeastern Louisiana University, Hammond, LA 70402-0687, USA Received 25 August 1999; accepted 3 September 1999 Submitted by B. Cain Dedicated to Hans Schneider Abstract We study the Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices, and consider some related results. © 1999 Elsevier Science Inc. All rights reserved. Keywords: Canonical form; Complex orthogonal matrix; Complex skew-symmetric matrix 1. Introduction and notation Every square complex matrix A is similar to its transpose, AT ([2, Section 3.2.3] or [1, Chapter XI, Theorem 5]), and the similarity class of the n-by-n complex symmetric matrices is all of Mn [2, Theorem 4.4.9], the set of n-by-n complex matrices. However, other natural similarity classes of matrices are non-trivial and can be characterized by simple conditions involving the Jordan Canonical Form. For example, A is similar to its complex conjugate, A (and hence also to its T adjoint, A∗ = A ), if and only if A is similar to a real matrix [2, Theorem 4.1.7]; the Jordan Canonical Form of such a matrix can contain only Jordan blocks with real eigenvalues and pairs of Jordan blocks of the form Jk(λ) ⊕ Jk(λ) for non-real λ.We denote by Jk(λ) the standard upper triangular k-by-k Jordan block with eigenvalue ∗ Corresponding author.
    [Show full text]