961ls03Matrix2 - Fall 2007 Linear Systems Chapter 03 Linear Algebra Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are

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Fall 2007 線性系統 Linear Systems Chapter 03 Linear Algebra Feng-Li Lian NTU-EE Sep07 – Jan08 Materials used in these lecture notes are adopted from “Linear System Theory & Design,” 3rd. Ed., by C.-T. Chen (1999) Outline ± Introduction ± Basis, Representation, & Orthonormalization (3.2) ± Linear Algebraic Equations (3.3) ± Similarity Transformation (3.4) ± Diagonal Form and Jordan Form (3.5) ± Functions of a Square Matrix (3.6) ± Lyapunov Equation (3.7) ± Some Useful Formulas (3.8) ± Quadratic Form and Positive Definiteness (3.9) ± Singular-Value Decomposition (3.10) ± Norms of Matrices (3.11)
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Matrix Algebra – 1 (3.1) Matrix partition and block matrix [] ij ⎡⎤ == ⎢⎥ ⎣⎦ AA ij BB NTUEE-LS1-Matrix-4 Feng-Li Lian © 2007 Matrix Algebra – 2 Matrix multiplication with compatible partitions = CA B Compatible Partitions : column partition of the 1st matrix = row partition of the 2nd matrix ik ij jk j = B =
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NTUEE-LS1-Matrix-5 Feng-Li Lian © 2007 Matrix Algebra – 3 Linear combination of A ’s columns Linear Dependence (L.D.) vs. Linear Independence (L.I.) – 1 (3.2)
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NTUEE-LS1-Matrix-7 Feng-Li Lian © 2007 Linear Dependence (L.D.) vs. Linear Independence (L.I.) – 2 A set of vectors in R n is a basis if 1) The set of vectors is L.I. ( linearly independent ) 2) All vectors in R n can be linearly combined by those in the set Every basis in R n has n vectors . Thus dim(R n ) = : n Let { x 1 , x 2 , …, x n } be a basis and x be a vector in R n , then x = [ x 1 x 2 xxx x n ] α Any set of more than n vectors in R n is L.D. unique representation of x w.r.t. the basis NTUEE-LS1-Matrix-8 Feng-Li Lian © 2007 Norm – 1 Standard orthonormal basis
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NTUEE-LS1-Matrix-9 Feng-Li Lian © 2007 Norm – 2 Norms of vectors : real-valued function (triangle inequality) satisfying NTUEE-LS1-Matrix-10 Feng-Li Lian © 2007 Norm – 3 Common norm: 1-norm , 2-norm ( Euclidean ), -norm, p-norm
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NTUEE-LS1-Matrix-11 Feng-Li Lian © 2007 Orthonormalization – 1 NTUEE-LS1-Matrix-12 Feng-Li Lian © 2007 Orthonormalization – 2 Schmidt orthonormalization procedure for { e 1 , e 2 , …, e m } :
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NTUEE-LS1-Matrix-13 Feng-Li Lian © 2007 Orthonormalization – 3 Representation by O rthonormal vectors { q 1 , q 2 , …, q m } : Q = [ q 1 q 2 xxx q m ] Ä Q Q = I m then α =[ q 1 q 2 xxx q n ] x and x = [ q 1 q 2 xxx q n ] α , Let { q 1 , q 2 , …, q n } be an orthonormal basis Ä Q = Q -1 Linear Algebraic Equations – 1 (3.3) A x = y , A : m × n , x : n × 1, y : m × 1 z range space of A : ={ β | β = A α } z rank( A ) : = ρ ( A ) : = dim. of A ’s range space min{ m , n } z null space of A : α | A α = 0 } z nullity( A ) : =d i m . o f A ’s null space = n ( A )
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NTUEE-LS1-Matrix-15 Feng-Li Lian © 2007 Linear Algebraic Equations – 2 z Ax = y has at least a solution ρ ( A ) = ( [ A y ] ) z Ax = y has at least a solution for every y ( A ) = m NTUEE-LS1-Matrix-16 Feng-Li Lian © 2007 Theorem 3.1
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NTUEE-LS1-Matrix-17 Feng-Li Lian © 2007 Linear Algebraic Equations – 3 z Suppose nullity( A ) = k , { n 1 , n 2 , …, n k } is a basis of the null space of A , and Ax = y has a solution x p z Then every vector x = x p + α 1 n 1 + 2 n 2 + xxx + k n k is also a solution NTUEE-LS1-Matrix-18 Feng-Li Lian © 2007 Theorem 3.2
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This note was uploaded on 07/21/2009 for the course EE 901-43400 taught by Professor Feng-lilian during the Fall '08 term at National Taiwan University.

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961ls03Matrix2 - Fall 2007 Linear Systems Chapter 03 Linear Algebra Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are

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