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Chapter 2: Matrix Algebra

Chapter 2: Matrix Algebra

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Chapter 2: Matrix Algebra
2.1 Matrix Addition, Scalar Multiplication, and Transposition
Example 2.1.1
Matrix Addition
Definition 2.1 Matrix Addition
Example 2.1.2
Example 2.1.3
Example 2.1.4
Solution.
Example 2.1.5
Scalar Multiplication
Definition 2.2 Matrix Scalar Multiplication
Example 2.1.6
Solution.
Example 2.1.7
Theorem 2.1.1
Example 2.1.8
Transpose of a Matrix
Definition 2.3 Transpose of a Matrix
Example 2.1.9
Solution.
Theorem 2.1.2
Example 2.1.10
Example 2.1.11
Example 2.1.12
Exercises for 2.1
2.2 Matrix-Vector Multiplication
Vectors
Definition 2.4 The set Rn of ordered n-tuples of real numbers
Matrix-Vector Multiplication
Example 2.2.1
Solution.
Definition 2.5 Matrix-Vector Multiplication
Theorem 2.2.1
Example 2.2.2
Example 2.2.3
Example 2.2.4
Example 2.2.5
Example 2.2.6
Theorem 2.2.2
Linear Equations
Theorem 2.2.3
Example 2.2.7
Theorem 2.2.4
The Dot Product
Definition 2.6 Dot Product in Rn
Theorem 2.2.5: Dot Product Rule
Example 2.2.8
Example 2.2.9
Example 2.2.10
Definition 2.7 The Identity Matrix
Example 2.2.11
Example 2.2.12
Theorem 2.2.6
Transformations
Example 2.2.13
Example 2.2.14
Definition 2.8 Matrix Transformation TA
Example 2.2.15
Solution.
Example 2.2.16
Example 2.2.17
Exercises for 2.2
2.3 Matrix Multiplication
Composition and Matrix Multiplication
Definition 2.9 Matrix Multiplication
Theorem 2.3.1
Example 2.3.1
Example 2.3.2
Theorem 2.3.2: Dot Product Rule
Compatibility Rule
Convention
Example 2.3.3
Example 2.3.4
Example 2.3.5
Example 2.3.6
Example 2.3.7
Theorem 2.3.3
Warning
Example 2.3.8
Example 2.3.9
Example 2.3.10
Example 2.3.11
Block Multiplication
Definition 2.10 Block Partition of a Matrix
Theorem 2.3.4: Block Multiplication
Theorem 2.3.5
Example 2.3.12
Directed Graphs
Theorem 2.3.6
Exercises for 2.3
Exercise 2.3.7
Exercise 2.3.8
Exercise 2.3.15
Exercise 2.3.22
Exercise 2.3.28
2.4 Matrix Inverses
Definition 2.11 Matrix Inverses
Example 2.4.1
Example 2.4.2
Theorem 2.4.1
Example 2.4.3
Example 2.4.4
Inverses and Linear Systems
Theorem 2.4.2
Example 2.4.5
An Inversion Method
Matrix Inversion Algorithm
Example 2.4.6
Theorem 2.4.3
Properties of Inverses
Example 2.4.7: Cancellation Laws
Example 2.4.8
Example 2.4.9
Theorem 2.4.4
Proof.
Corollary 2.4.1
Example 2.4.10
Theorem 2.4.5: Inverse Theorem
Corollary 2.4.1
Corollary 2.4.2
Example 2.4.11
Inverses of Matrix Transformations
Theorem 2.4.6
Example 2.4.12
Exercises for 2.4
Exercise 2.4.8
Exercise 2.4.10
Exercise 2.4.38
2.5 Elementary Matrices
Definition 2.12 Elementary Matrices
Lemma 2.5.1: 10
Lemma 2.5.2
Example 2.5.1
Inverses and Elementary Matrices
Theorem 2.5.1
Example 2.5.2
Theorem 2.5.2
Example 2.5.3
Smith Normal Form
Theorem 2.5.3
Example 2.5.4
Uniqueness of the Reduced Row-echelon Form
Theorem 2.5.4
Exercises for 2.5
Exercise 2.5.5
2.6 Linear Transformations
Linear Transformations
Definition 2.13 Linear Transformations Rn Rm
Theorem 2.6.1: Linearity Theorem
Example 2.6.1
Example 2.6.2
Theorem 2.6.2
Example 2.6.3
Example 2.6.4
Example 2.6.5
Theorem 2.6.3
Example 2.6.6
Some Geometry
Scalar Multiple Law
Parallelogram Law
Rotations
Theorem 2.6.4
Example 2.6.7
Reflections
Theorem 2.6.5
Example 2.6.8
Projections
Theorem 2.6.6
Example 2.6.9
Exercises for 2.6
2.7 LU-Factorization15
Triangular Matrices
Example 2.7.1
Lemma 2.7.1
LU-Factorization
Theorem 2.7.1
Definition 2.14 LU-factorization
Example 2.7.2
LU-Algorithm
Example 2.7.3
Example 2.7.4
Theorem 2.7.2
Example 2.7.5
Theorem 2.7.3
Corollary 2.7.1
Proofs of Theorems
Lemma 2.7.2
Exercises for 2.7
Exercise 2.7.6
2.8 An Application to Input-Output Economic Models16
Example 2.8.1
Theorem 2.8.1
Example 2.8.2
The Open Model
Theorem 2.8.2
Heuristic Proof.
Example 2.8.3
Corollary 2.8.1
Exercises for 2.8
2.9 An Application to Markov Chains
Definition 2.15 Markov Chain
Example 2.9.1
Example 2.9.2
Theorem 2.9.1
Example 2.9.3
Steady State Vector
Theorem 2.9.2
Example 2.9.4
Exercises for 2.9
Supplementary Exercises for Chapter 2
Exercise 2.5
Exercise 2.10
Section worksheets