Definitions and Theorems 3

Thursday, April 12, 2012

1:39 PM

 Chapter 5

 

Right Inverse
Left inverse

 

Theorem 5.1.1



 

Corollary 5.1.2



 

Theorem 5.1.3



 

Matrix Inverse
Invertible

 

Theorem 5.1.4



 

Theorem 5.1.5



1. 
2. 
3. 

 

Theorem 5.1.6



 

Theorem 5.1.7 (Invertible Matrix Theorem)



1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 

 

 

 

 

Elementary Matrix

 

Theorem 5.2.1.



 

Theorem 5.2.2



 

Theorem 5.2.3



 

Theorem 5.2.4



 

Theorem 5.2.5





 

Corollary 5.2.6



 

Theorem 5.2.7



 

 

Cofactor

 

Theorem 5.3.1











 

Upper Triangular Lower triangular

 

Theorem 5.3.2





 

Theorem 5.3.3



 

Theorem 5.3.4



 

Corollary 5.3.5



 

Theorem 5.3.6



 

Corollary 5.3.7





 

Theorem 5.3.8 (Addition to the Invertible Matrix Theorem)



 

Theorem 5.3.9



 

Corollary 5.3.10



 

Theorem  5.3.11



 

 

 

Lemma 5.4.1





 

Theorem 5.4.2



 

Cofactor Matrix
Adjugate

 

Theorem 5.4.3 (Cramer's Rule)







 

 

 

Area of a parallelogram
Height of Parallelepiped
Volume of Parallelepiped

 

 

 

 

 Chapter 6 - Diagonalization

 

	It satisfies
Diagonal Matrix

 

Theorem 6.1.1



Then

1. 
2. 
3. 

 

Trace

 

Similar Matrices

 

 

 



Eigenvalue Eigenvector Eigenpair
Eigenvalues Eigenvectors

 

Characteristic Polynomial

 

Theorem 6.2.1



 

Eigenspace

 

Algebraic Multiplicity Geometric Multiplicity

 

Lemma 6.2.2



 

Theorem 6.2.3





 

 

 

Diagonalizable

 

Theorem 6.3.1



 

Theorem 6.3.2



 

Corollary 6.3.3



 

Deficient

 

Corollary 6.3.4



 

Theorem 6.3.5



Then



 

 

Theorem 6.4.1