What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010

Transcription

1 What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 Scott Strong Colorado School of Mines What is the Matrix? p. 1/20

2 Overview/Keywords/References Advanced Engineering Mathematics Slide Set Linear Algebra Introduction Matrices and their Algebra Reference Text: EK 7.1,7.2 See Also: Lecture Notes : 01.Linear Definitions Lecture Notes : 02.Introduction to Linear Equations What is the Matrix? p. 2/20

3 Before We Begin Quote of Slide Set Up and down, and in the end it s only round and round and round. Pink Floyd : Us and Them (1973) What is the Matrix? p. 3/20

4 Introduction Question: So, what now? Answer: Recall that we started the semester by asking the question, given the linear transformation T and inhomogeneity y can you find x such that, This is the fundamental linear problem. T(x) = y. (1) Key Point: If you can find x it must take the form, x = x h + x p, (2) where x h is the solution to (1) where y = 0 and x p is a particular solution to (1). What is the Matrix? p. 4/20

5 Linear Transformations : Recall The fact that the solution to (1) must take the form (2) is the fundamental point of linear theory. That is, The general solution to a linear problem is nothing more than the sum of its parts. Question: What must be true for T to be linear? Answer: For T to be a linear transform we must have that, Preservation of Origins: T(0) = 0 Transformations of Linear Combinations to Combinations of Transformations: T(αx + βy) = αt(x) + βt(y) Example: T = [ t ] 2 c2 x 2 T(u) = u t c2 2 u x 2 (3) What is the Matrix? p. 5/20

6 The Heat Equation : Redux At this point we clearly see that our example (3) defines a heat equation. That is, T(u) = F u t = u c2 2 + F. (4) x2 We found that the solution to the corresponding homogeneous problem, T(u) = 0, is given by, u(x,t) = a 0 + n=1 ( ) ( ) a n cos λn x e c2 λ n t + b n sin λn x e c2 λ n t, where a 0,a n,b n,λ n depend on the boundary and initial conditions for the problem. (5) What is the Matrix? p. 6/20

7 Linear Transformations : Redux Remember that the fundamental solution to T(x) = 0 was called the null-space or kernel of the transform T and defines the underlying vector-space that is being transformed by T. That being said, Key Point 1: The linear transform (3) defines a kernel, which is a vector-space whose elements/points are solutions to the homogeneous heat equation. Key Point 2: To get to points in this vector-space we use linear combinations (Fourier series) of a set of basis vectors (Fourier modes). Key Point 3: Since the total number of basis vectors defines a space s dimension, we conclude that the solution space to the heat equation has infinite dimension. What is the Matrix? p. 7/20

8 Linear Transformations : Simplifications Question: What if the underlying space transformed by T has finite dimension? Answer: Then T is called a matrix A and the study of linear transformations of finite-dimensional spaces is called linear algebra. Linear algebra is an old topic on which there is much to say and know. See also: MATH332 - Linear Algebra. We will concentrate on the following: 1. The Algebra of Matrices 2. Solutions to Linear Systems via Row-Reduction 3. Square Systems and Properties of Invertible Matrices 4. Linear Scalings and Eigenproblems 5. Matrix Diagonalization What is the Matrix? p. 8/20

9 Linear Algebra - Cast Overview Before we present the formal definitions for the mathematical objects of linear algebra we make the following points: Matrix : A matrix is a mathematical spreadsheet that turns out to be a convenient way to organize data. Unary Operations : A unary operation takes a matrix and operate on it producing a new matrix. Binary Operations : A binary operation takes two matrices and operates with them producing a single new matrix. What is the Matrix? p. 9/20

10 Matrix - Definition Definition: A matrix is a set of elements organized by two indices into a rectangular array. In the case that these objects exist in the set of real numbers we write A R m n, where n,m N. a 11 a 12 a a 1n A = a 21 a 22 a a 2n a 31 a 32 a a 3n a m1 a m2 a m3... a mn, a ij R, i = 1,2,3, m j = 1,2,3,,n. (6) What is the Matrix? p. 10/20

11 Matrix - Notes Notes: Typically, a ij are numbers but they can be any well-defined mathematical object. Often elements are written [A] ij = a ij. A vector is a special case of a matrix where n = 1. A scalar is a special case of a matrix where m = n = 1. Elements a ii are called main-diagonal elements of A. If m = n then the matrix is called square. Two matrices are said to be equal if they are the same dimension and equal at the element level. That is, A m n = B p q if and only if m = p and n = q and a ij = b ij for i = 1,2,3,...,m and j = 1,2,3,...,n. What is the Matrix? p. 11/20

12 Transposition Definition: Given A R m n we define the transpose of A to be the matrix A T R n m, such that: a 11 a 21 a a m1 a 12 a 22 a a m2 A T = a 13 a 23 a a m3, [ A T]. ij = a i = 1,2,3, m ji,..... j = 1,2,3,,n. a 1n a 2n a 3n... a mn If A is such that A= A T then the matrix A is called symmetric. If A is such that A T = A then the matrix A is called skew-symmetric. (7) What is the Matrix? p. 12/20

13 Conjugation Definition: Given A C m n, define the conjugate of A to be the matrix Ā Cm n such that, ā 11 ā 12 ā ā 1n ā 21 ā 22 ā ā 2n Ā = ā 31 ā 32 ā ā 3n ā m1 ā m2 ā m3... ā mn. (8) The bar implies complex conjugation. That is if c C then c = a + bi, a,b R and c = a bi. What is the Matrix? p. 13/20

14 Hermitian Adjoint Definition: Given A C m n, define the adjoint or Hermitian of A to be the matrix A H C m n such that A H = (Ā)T = ( A T). It is often the case that the Hermitian is denoted A. The adjoint is considered as an extension of the transpose to matrices with complex numbers. A matrix is called self-adjoint if A H = A. A matrix is called skew-adjoint if A H = A. What is the Matrix? p. 14/20

15 The Vector Space Structure We say V is a vector space if there are two binary operations, + and, such that, Commutativity : a + b = b + a, a, b V Associativity : a + (b + c) = (a + b) + c, a, b, c V Additive Identity : a + 0 = a for every a V Additive Inverse : a + ( a) = 0 for every a V Distribution: (α + β)(a + b) = (α + β)a + (α +β)b,α,β R Associativity : Let α,β R then α(βa) = (αβ)a,, a V Multiplicative Identity : 1a = a Matrix Addition: A + B = C, c ij = a ij + b ij, A, B, C R m n Scalar Multiplication: [αa] ij = αa ij, α R, A R m n The space of matrices R m n is a vector space. What is the Matrix? p. 15/20

16 Matrix Product - Definition Definition: Let A R m n and B R n q then AB = C R m q is defined and has elements, n c ij = a ik b kj. (9) k=1 Properties: 1. A(BC) = (AB)C 2. A(B + C) = AB + AC 3. (B + C)A = BA + CA 4. r(ab) = r(a)b = ArB, where r R 5. I m A = A = AI n where the identity I k is such that [I k ] ij = δ ij. 6. (AB) T = B T A T What is the Matrix? p. 16/20

17 Matrix Product - Notes Note for the product to be defined the number of columns in the matrix on the left must be equal to the number of row in the matrix on the right. The resulting element c ij is the dot-product of the i th row of A with the j th column of B. An important consequence of this definition of matrix product is that it does not generally commute. That is AB can be different than BA. Another important consequence of this definition is that the problem Ax = b where A R m n, x R n and b R m corresponds to m many linear equations of n many unknowns. What is the Matrix? p. 17/20

18 Example 1 In the following example we show how to apply the definition of matrix multiplication and as a consequence show that matrices will not generally commute. Example: Let A = [ ], B = , (10) and note that both AB and BA are both defined. However, [ ] AB =, BA = (11) What is the Matrix? p. 18/20

19 Example 2 We now note the importance of defining matrix multiplication through the standard dot-product. Consider, a 11 a 12 a a 1n x 1 b 1 a 21 a 22 a a 2n x 2 b 2 A = a 31 a 32 a a 3n, x = x 3, b = b , a m1 a m2 a m3... a mn x n b n which is m many linear equations of n many unknowns a 11 x 1 + a 12 x 2 + a 13 x a 1n x n = b 1. a m1 x 1 + a m2 x 2 + a m3 x a mn x n = b m What is the Matrix? p. 19/20