# Orthogonal Diagonalization of Symmetric Matrices

Size: px
Start display at page:

Transcription

1 MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding a new, orthogonal, basis v 1,..., v k by setting v 1 = u 1. Then we seek v 2 = u 2 + c 1 u 1 with unknown coeff. c 1 ; we need v 2 v 1 = 0, that is, u 2 u 1 + c 1 ( u 1 u 1 ) = 0, whence c 1 = u 2 u 1 u 1 u 1 (note that u 1 u 1 0 as u 1 0 being a vector in a basis). Next we seek v 3 = u 3 + d 1 v 1 + d 2 v 2 with unknown coeff. d 1, d 2 ; we need v 3 v 1 = 0, that is, u 3 v 1 + c 1 ( v 1 v 1 ) = 0 (the term with v 2 v 1 = 0 vanishes), whence d 1 = u 3 v 1 v 1 v 1 ; also we need v 3 v 2 = 0, that is, u 3 v 2 + c 1 ( v 2 v 2 ) = 0 (the term with v 1 v 2 = 0 vanishes), whence d 2 = u 3 v 2 v 2 v 2....And so on, at each step we seek v j = u j + linear comb. of preceding v 1,..., v j 1 with unknown coeff., which are easily found. Orthogonal Diagonalization of Symmetric Matrices Definition. A square matrix A is orthogonally diagonalizable if there exists an orthogonal matrix Q such that Q T AQ = D is a diagonal matrix. Remarks. Since Q T = Q 1 for orthogonal Q, the equality Q T AQ = D is the same as Q 1 AQ = D, so A D, so this a special case of diagonalization: the diagonal entries of D are eigenvalues of A, and the columns of Q are corresponding eigenvectors. The only difference is the additional requirement that Q be orthogonal, which is equivalent to the fact that those eigenvectors columns of Q form an orthonormal basis of R n. If a matrix is orthogonally diagonalizable, then it is sym- Theorem metric. Proof. We have Q T AQ = D; times Q on the left, and Q T on the right gives A = QDQ T (since Q T = Q 1 ). Then A T = (QDQ T ) T = (Q T ) T D T Q T = QDQ T = A, so A is symmetric. Theorem All eigenvalues (all roots of the characteristic polynomial) of a symmetric matrix are real. Theorem Eigenvectors of a symmetric matrix corresponding to different eigenvalues are orthogonal. Proof. Let A T = A have eigenvectors v 1 and v 2 for eigenvalues λ 1 λ 2. We compute the dot product (A v 1 ) v 2 = (λ 1 v 1 ) v 2 = λ 1 ( v 1 v 2 ). On the other hand, the left-hand side can be written as a matrix product: (A v 1 ) v 2 =

2 MATH10212 Linear Algebra Brief lecture notes 58 (A v 1 ) T v 2 = v T 1 A T v 2 = v T 1 (A v 2 ) = v 1 (λ 2 v 2 ) = λ 2 ( v 1 v 2 ). Thus, λ 1 ( v 1 v 2 ) = λ 2 ( v 1 v 2 ). Since λ 1 λ 2 by hypothesis, we must have v 1 v 2 = 0. Theorem Every symmetric matrix is orthogonally diagonalizable. Method for orthogonal diagonalization of a symmetric matrix. Find eigenvalues of A. Find the eigenspace for each eigenvalue. For repeated eigenvalues (when the dimension of the eigenspace is greater than 1) apply Gram Schmidt orthogonalization to find an orthogonal basis. Together, these orthogonal bases of eigenspaces form an orthogonal basis of R n. Normalize, dividing each vector of the basis by its length. The resulting orthonormal basis can be taken as columns of Q such that Q T AQ = Q 1 AQ = D, where D is diagonal with the eigenvalues of A on the diagonal, in the order corresponding to the order of their eigenvectors as columns of Q. Example. For A = the characteristic polynomial is λ 2 2 det(a λi) = 2 1 λ λ = (1 λ) (1 λ) 4(1 λ) 4(1 λ) = = (λ 5)(λ + 1) 2. Thus, eigenvalues are 5 and 1. Eigenspace E 1 : (A ( 1)I) x = 0; x x 2 = 0 0 ; x 1 = x 2 x 3, x 3 0 s t where x 2, x 3 are free var.; E 1 = s s, t R ; t a basis of E 1 : u 1 = 1 1, u 2 = Apply Gram Schmidt to orthogonalize: v 1 = u 1 ; seek v 2 = u 2 + c v 1 ; for v 2 v 1 = 0 obtain c = u2 v1 v 1 v 1 = 1 2, thus, v 2 = u 2 (1/2) v 1 = 1/2 1/2. 1 Eigenspace E 5 : (A 5I) x = 0; x x 2 = 0 0 ; solve this x 3 0 system...: x 1 = x 2 = x 3, where x 3 is a free var.; E 5 = t t t R ; t a basis of E 5 : (Note, E 5 is automatically orthogonal to E 1.)

3 MATH10212 Linear Algebra Brief lecture notes 59 Together, we have an orthogonal basis of R 3 consisting of eigenvectors: 1 1, 1/2, Normalize: 1/ 2 1/ 1/ 6 2, 1/ 1/ 3 6, 1/ 3 0 2/3 1/. 3 1/ 2 1/ 6 1/ 3 Let Q = 1/ 2 1/ 6 1/ 3, which is an orthogonal matrix; 0 2/3 1/ 3 then Q T AQ = Orthogonal Complements and Orthogonal Projections Definition Let W be a subspace of R n. We say that a vector v in R n is orthogonal to W if v is orthogonal to every vector in W. The set of all vectors that are orthogonal to W is called the orthogonal complement of W, denoted W. That is, W = { v in R n v w = 0 for all w in W } Theorem 5.9 Let W be a subspace of R n. a. W is a subspace of R n. b. (W ) = W. c. W W = { 0}. d. If W = span( w 1,..., w k ), then v is in W if and only if v w i = 0 for all i = 1,..., k. Theorem 5.10 Let A be an m n matrix. Then the orthogonal complement of the row space of A is the null space of A, and the orthogonal complement of the column space of A is the null space of A T : (row(a)) =null(a) and (col(a)) =null(a T )

4 MATH10212 Linear Algebra Brief lecture notes 60 Orthogonal Projections Theorem 5.11 The Orthogonal Decomposition Theorem Let V be a subspace of R n and let u be a vector in R n. Then there are unique vectors v V and w V such that u = v + w. In fact: if v 1,..., v r is an orthogonal basis of V, then an orthogonal basis v r+1,..., v n of V must have n r vectors, and together these n vectors form an orthogonal basis v 1,..., v r, v r+1,..., v n of R n Definition In Theorem 5.11, the vector v is called the orthogonal projection of u onto V, and the length of w is called the distance from u to V. Method for finding the orthogonal projection and the distance a subspace V of R n and some vector u R n ). (given Choose an orthogonal basis v 1,..., v r of V (if V is given as a span of some non-orthogonal vectors, apply Gram Schmidt first to obtain an orthogonal basis of V ); we know that there is an orthogonal basis v r+1,..., v n of V such that v 1,..., v r, v r+1,..., v n is an orthogonal basis of R n (but we do not really need these v r+1,..., v n!). Then u = r a i v i + i=1 n j=r+1 b j v j. We now find the coefficients a i. For that, take dot product with v i0 for each i 0 = 1,..., r: on the right only one term does not vanish, since the v k are orthogonal to each other: u v i0 = a i0 ( v i0 v i0 ), whence a i0 = u v i 0 v i0 v i0. Having found these a i0 for each i 0 = 1,..., r, we now have the orthogonal projection of u onto V : r v = a i v i. i=1 The orthogonal component is found by subtracting: w = u v V, and the distance from u to V is w.

5 MATH10212 Linear Algebra Brief lecture notes 61 Corollary 5.12 If W is a subspace of R n, then (W ) = W Theorem 5.13 If W is a subspace of R n, then dim W + dim W = n Corollary 5.14 The Rank Theorem If A is an m n matrix, then rank (A) + nullity (A) = n Vector Spaces and Subspaces Definition Let V be a set on which two operations, called addition and scalar multiplication, have been defined. If u and v are defined in V, the sum of u and v is denoted by u + v, and if c is a scalar, the scalar multiple of u is denoted by c u. If the following axioms hold for all u, v and w in V and for all scalars c and d, then V is called a vector space and its elements are called vectors. 1. u + v is in V. Closure under addition 2. u + v = v + u Commutativity 3. ( u + v) + w = u + ( v + w) Associativity 4. There exists an element 0 in V, called a zero vector such that u+ 0 = u. 5. For each u in V, there is an element u in V such that u + ( u) = c u is in V. Closure under scalar multiplication 7. c( u + v) = c u + c v Distributivity 8. (c + d) u = c u + d u Distributivity 9. c(d u) = (cd) u u = u

6 MATH10212 Linear Algebra Brief lecture notes 62 Theorem 6.1 Let V be a vector space, u a vector in V, and c a scalar. a. 0 u = 0 b. c 0 = 0 c. ( 1) u = u d. If c u = 0, then c = 0 or u = 0. Subspaces Definition A subset W of a vector space V is called a subspace of V if W is itself a vector space with the same scalars, addition and scalar multiplication as V. Theorem 6.2 Let V be a vector space and let W be a nonempty subset of V. Then W is a subspace of V if and only if the following conditions hold: a. If u and v are in W, then u + v is in W. b. If u is in W and c is a scalar, then c u is in W. Example 6.14 If V is a vector space, then V is clearly a subspace of itself. The set { 0}, consisting of only the zero vector, is also a subspace of V, called the zero subspace. To show this, we simply note that the two closure conditions of Theorem 6.2 are satisfied: = 0 and c 0 = 0 for any scalar c The subspaces { 0} and V are called the trivial subspaces of V. Spanning Sets Definition If S = { v 1, v 2,..., v k } is a set of vectors in a vector space V, then the set of all linear combinations of v 1, v 2,..., v k is called the span of and is denoted by v 1, v 2,..., v k span( v 1, v 2,..., v k ) or span(s). If V =span(s), then S is called a spanning set for V and V is said to be spanned by S.

7 MATH10212 Linear Algebra Brief lecture notes 63 Theorem 6.3 Let v 1, v 2,..., v k be vectors in a vector space V. a. span( v 1, v 2,..., v k ) is a subspace of V. b. span( v 1, v 2,..., v k ) is a smallest subspace of V that contains v 1, v 2,..., v k. Linear Independence, Basis and Dimension Linear Independence Definition A set of vectors S = { v 1, v 2,..., v k } in a vector space V is linearly dependent if there are scalars c 1, c 2,..., c k at least one of which is not zero, such that c 1 v 1 + c 2 v c k v k = 0 A set of vectors that is not linearly dependent is said to be linearly independent. Theorem 6.4 A set of vectors S = { v 1, v 2,..., v k } in a vector space V is linearly dependent if and only if at least one of the vectors can be expressed as a linear combination of the others. Bases Definition A subset B of a vector space V is a basis for V if 1. B spans V and 2. B is linearly independent.

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### Linear Algebra Review. Vectors

Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

### 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

### Applied Linear Algebra I Review page 1

Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

### Inner product. Definition of inner product

Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

### 4 MT210 Notebook 4 3. 4.1 Eigenvalues and Eigenvectors... 3. 4.1.1 Definitions; Graphical Illustrations... 3

MT Notebook Fall / prepared by Professor Jenny Baglivo c Copyright 9 by Jenny A. Baglivo. All Rights Reserved. Contents MT Notebook. Eigenvalues and Eigenvectors................................... Definitions;

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### 1 Introduction to Matrices

1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### Linearly Independent Sets and Linearly Dependent Sets

These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,

### Examination paper for TMA4115 Matematikk 3

Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round \$200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### [1] Diagonal factorization

8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3

Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

### MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α

### Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

### Subspaces of R n LECTURE 7. 1. Subspaces

LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r

### 1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

### Lecture 5: Singular Value Decomposition SVD (1)

EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system

### Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### October 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix

Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Numerical Methods I Eigenvalue Problems

Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)

### MA106 Linear Algebra lecture notes

MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Orthogonal Projections and Orthonormal Bases

CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).

### Inner Product Spaces. 7.1 Inner Products

7 Inner Product Spaces 71 Inner Products Recall that if z is a complex number, then z denotes the conjugate of z, Re(z) denotes the real part of z, and Im(z) denotes the imaginary part of z By definition,

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

### Nonlinear Iterative Partial Least Squares Method

Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for

### Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### 9 Multiplication of Vectors: The Scalar or Dot Product

Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

### A note on companion matrices

Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod

### SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated

### SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

### Linear Algebra: Determinants, Inverses, Rank

D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Inner products on R n, and more

Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

### MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

### Lecture Notes 2: Matrices as Systems of Linear Equations

2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

### CS3220 Lecture Notes: QR factorization and orthogonal transformations

CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss

### 1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

### Brief Introduction to Vectors and Matrices

CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vector-valued

### Lecture notes on linear algebra

Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra

### Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but