Help me understand a line in an ATA is positive, semi-definite proof

  • Context: Graduate 
  • Thread starter Thread starter psholder
  • Start date Start date
  • Tags Tags
    Line Positive Proof
Click For Summary
SUMMARY

The discussion centers on the proof that the expression ATA is positive semi-definite when A is an n x n matrix. The key line in question, vTATA v = ATv • ATv, demonstrates that vTATA v is nonnegative. This equivalence is established through the relationship (ATv)T ATv = (vTA) ATv, clarifying the connection between the two forms. Understanding this proof requires a solid grasp of linear algebra concepts.

PREREQUISITES
  • Linear algebra fundamentals, specifically matrix multiplication.
  • Understanding of positive semi-definite matrices.
  • Familiarity with vector and matrix transposition.
  • Knowledge of inner product notation and properties.
NEXT STEPS
  • Study the properties of positive semi-definite matrices in linear algebra.
  • Learn about matrix transposition and its implications in proofs.
  • Explore the concept of inner products and their geometric interpretations.
  • Investigate the implications of the Spectral Theorem for symmetric matrices.
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, particularly those studying linear algebra, as well as anyone involved in theoretical computer science or data science who requires a deeper understanding of matrix properties.

psholder
Messages
1
Reaction score
0
help me understand a line in an "ATA is positive, semi-definite" proof

I am looking at a proof for why ATA is positive semi-definite when A is nxn and it has this line.

vTAATv = ATv • ATv ≥ 0.

I understand what vTAATv means and the purpose of proving that it's nonnegative, etc... My problem is that I am a linear algebra novice and do not necessarily understand how the first part vTAATv is equivalent to ATv • ATv. I know that aTb = a • b, but something else is going on, no? Appreciate any help!
 
Physics news on Phys.org
welcome to pf!

hi psholder! welcome to pf! :smile:

(try using the X2 button just above the Reply box :wink:)
psholder said:
… I am a linear algebra novice and do not necessarily understand how the first part vTAATv is equivalent to ATv • ATv. I know that aTb = a • b, but something else is going on, no? Appreciate any help!

ATv • ATv

= (ATv)T ATv

= (vTA) ATv :wink:
 

Similar threads

Undergrad How can I test for positive semi-definiteness in matrices?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Proof that covariance matrix is positive semidefinite
  • · Replies 1 ·
Replies
1
Views
9K
Undergrad Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r
  • · Replies 4 ·
Replies
4
Views
2K
MHB Understanding Bland's Proof of Proposition 4.3.14: Primitive Elements of Modules
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Proof that there exists a smallest positive linear combination?
  • · Replies 1 ·
Replies
1
Views
6K
Help with Time-Independent Perturbation Theory "Good" States Proof
  • · Replies 4 ·
Replies
4
Views
944
Undergrad Help Needed: Understanding Hungerford's Algebra Book Proofs
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad K-algebra/associative algebra - equivalence of definitions
  • · Replies 7 ·
Replies
7
Views
4K
MHB Equivalence of Definitions of Algebras in D&F and Cooperstein | Peter
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Simple Proofs for Matrix Algebra Properties: A Beginner's Guide
  • · Replies 3 ·
Replies
3
Views
3K