QR Fatoration (Linear Algebra) (FIXED)

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SUMMARY

The discussion centers on the QR factorization of a matrix A with linearly independent columns, expressed as A = QR, where Q is an orthonormal matrix and R is an upper triangular matrix. The key points include the necessity for R to be upper triangular due to the orthogonality of the columns in Q, and the requirement for the diagonal elements of R to be positive, which stems from the properties of the Gram-Schmidt process used in deriving the factorization. The participant seeks clarification on proving the orthogonality of vectors and the positivity of the diagonal elements.

PREREQUISITES
  • Understanding of QR factorization in linear algebra
  • Familiarity with orthonormal vectors and their properties
  • Knowledge of the Gram-Schmidt process
  • Basic concepts of matrix theory, particularly upper triangular matrices
NEXT STEPS
  • Study the Gram-Schmidt process in detail to understand how it generates orthonormal vectors
  • Learn about the properties of upper triangular matrices and their implications in linear algebra
  • Research proofs regarding the positivity of diagonal elements in upper triangular matrices
  • Explore applications of QR factorization in numerical methods and data analysis
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as data scientists and engineers utilizing QR factorization in computational applications.

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Homework Statement



Suppose that A is an matrix with linearly independent columns then A can be factored as,
where Q is an matrix with orthonormal columns and R is an invertible upper triangular matrix.

Show why it is Upper triangular matrix, and why the elements of the diagonal are Positive.

Homework Equations



A=QR
A= [x1 x2 ... xk]
R= [r1 r2 ... rk]

The Attempt at a Solution



(Sorry for my bad English, I am from Brazil)x1= (x1.u1)u1+(x1.u2)u2+...+(x1.uk)uk
x2= (x2.u1)u1+(x2.u2)u2+...+(x2.uk)uk
.
.
.
xn= (xn.u1)u1+(xn.u2)u2+...+(xn.uk)uk

now there is an argument, given Xi and Uj, if i<j , Uj and Xi are orthogonals. Which I do not know how to prove it.
So, if this is valid:

x1= (x1.u1)u1
x2= (x2.u1)u1+(x2.u2)u2
.
.
.
xk= (xk.u1)u1+(xk.u2)u2+...+(xk.uk)ukwhich is triangular. Now I do not now why the product (x1.u1), (xk.uk) is always positive.
 
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