Linear Operators

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Linear Operators Over Vector Spaces
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Desired Contents

• Linear Spaces over a Field
• Linear Spaces Bases and Representations
• Linear Operators and Their Representations
• Systems of Linear Algebraic Equations
• Diagonalization of Linear Operators
• Functions of a Square Matrix
• Norms and Inner Product

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Vector Spaces

• Definition of Fields
• Definition of a Vectors Space over a Field
• Examples of Vector Spaces and Subspaces
• Linear Independence Basis and Representations
• Change of Basis
• Linear Operators and Their representation
• Matrix representation of a linear operator
• Matrix Algebra

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Matrix Definition
Let Mm,n Rmxn Array of numbers
arranged in rows and columns A? Mm,n matrix m
rows n columns aij are its elements
column
row
Also denoted A aij Rank of A the of its
linearly independent rows (colums) Square matrix
if m n. Otherwise rectangular Row/column
vector when m 1 and n 1, respectively Submatr
ix of A the matrix obtained removing some
rows/columns
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Special Matrices
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Special Matrices
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Special Matrices
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Addition
Definition if A,B?Mm,n (have same dimension)
C A B where cij aij bij
(Mm,n, , 0, -(.)) is a commutative GROUP
?A,B?Mm,n AB ? Mm,n closure A B
B A commutative A
(B C) (A B) C associative A
0 0 A A identity A
(-A) (-A) A 0 inverse
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Multiplication
(Mm,n, x, 0, (.)-1) is a GROUP
?A?Mm,n,B?Mn,p AB ? Mm,p closure A(BC)
(AB)C associative AI IA
A identity A
(A-1) (A-1) A I inverse AB
? BA generally,
not commutative
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Matrix Multiplication (Justification)
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Matrix Multiplication (Justification)
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Matrix Multiplication (Justification)
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Matrix Multiplication (Justification)
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Division Ring
DR (Mn,n, , x, 0, I, (-.), (.)-1) is a
DIVISION RING (Mn,n, , 0, -(.))
commutative group (Mn,n,
x, I, (.)-1) group
A x (B C) (A x B) (A x C)
distributivity 1 (A B) x C (A x C) (A
x B) distributivity 2 However A (B x
C) ? (A B) x (A C) (holds if
A,B,C?Bool!)
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Vector Space
vectors
scalars
V (Mm,n,R,?) is a VECTOR SPACE Define
aA aaij Multiplication with a
Scalar (Mm,n, , 0, -(.))
commutative group a (B C) (aB) (aC)
distributivity of ? over vector (a b) C
(aC) (bC) distributivity of ? over scalar
a(bA) (ab)A
compatibility of multiplications 1A A1
A scalar identity Note ? is
ommited in text
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Transposition
Properties of transposition A
AT if A is symmetric (AB)T
BTAT product transposition (A
B)T AT BT sum transposition
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Determinants
cofactor
minor
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Properties of Determinants
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Properties of Determinants
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Adjugate
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Inverse
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Matrix Diagonalization
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Question

• A linear operator f has many representations
  f?
• Is there a basis ? in which f? is nice and
  simple?
• Simplest is a diagonal matrix diag(?1,,?n)

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Eigenvalues and Eigenvectors
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Eigenvalues and Eigenvectors
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Characteristic Polynomial (CP)
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Example Eigenvalues of A
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Example Eigenvalues of A
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Example Eigenvectors of A
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Case 1 All Eigenvalues of A are Distinct
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Independence of Eigenvectors
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Independence of Eigenvectors
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Diagonalization of A
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Diagonalization of A
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Diagonalization of A
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Diagonalization of A
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Example Diagonalization of A
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Example Diagonalization of A
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Example Diagonalization of A
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Example Diagonalization of A
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Case 2 Eigenvalues of A Have Multiplicity
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Degenerate Characteristic Polynomial
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Diagonalization of A
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Example Diagonalization of A
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Example Diagonalization of A
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Example Diagonalization of A
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Example Diagonalization of A
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Jordan Canonical Form
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Variants of Jordan Canonical Form
Form depends on the characteristics of A
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Generalized Eigenvector of Grade k for A
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Chain of Generalized Eigenvectors of length k
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Properties Generalized Eigenvectors Chains
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Properties Generalized Eigenvectors Chains
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Computing the Canonical Jordan Form
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Independence of Generalized Eigenvectors
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Independence of Generalized Eigenvectors
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Independence of Generalized Eigenvectors
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Independence of Generalized Eigenvectors
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Independence of Generalized Eigenvectors
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Representation of A in Jordan Canonical Form
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Homework
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Functions of a Square Matrix
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Matrix Polynomials
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Matrix Exponentials
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Jordan Matrix Powers
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Jordan Blocks Polynomials
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Jordan Blocks Polynomials
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Jordan Blocks Time Powers
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Jordan Canonical Form Polynomials
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Jordan Blocks Exponentials
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General Time Powers of Matrices
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General Time Exponentials of Matrices