Matric Proof: A, X', and Inverse - All You Need to Know | AA=A

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SUMMARY

The discussion centers on the matrix equation A = I - X(X'X)⁻¹X', where I represents the identity matrix. Key questions include whether matrices A, X, and (X'X) must be square matrices and whether A is idempotent (i.e., AA = A). The consensus is that A and (X'X) must be square matrices, while X does not necessarily have to be square. The discussion emphasizes the importance of understanding matrix properties and operations in linear algebra.

PREREQUISITES
  • Understanding of matrix operations, specifically transposition and inversion.
  • Familiarity with identity matrices and their properties.
  • Knowledge of idempotent matrices and their characteristics.
  • Basic concepts of linear algebra, particularly regarding square matrices.
NEXT STEPS
  • Study the properties of identity matrices in linear algebra.
  • Learn about matrix inversion and conditions for invertibility.
  • Research idempotent matrices and their applications in statistics.
  • Explore the implications of matrix dimensions in linear transformations.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as data scientists and statisticians working with matrix computations.

BigDave48
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Let A= I - X(X'X)inverseX' to clarify: '=transpose; inverse of quantity (X'X)

1. Must A be a square matrix?
2. Must (X'X) be a square matrix?
3. Must X be a square matrix?
4. Show whether matrix A is an idempotent matrix (i.e. that AA=A)

Thanks.
 
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im so stupid! i can't follow aNY OF THAT. if i could understand the question i could answer it instantly.
 
Sorry. I really would appreciate some help.

To be more clear: I is an identity matrix

Questions 1,2,and 3 are essentially Yes/No with a provided proof or explanation.

Does this help, clarify?

Thanks.
 
come on man.
 
What do you mean by
"Let A= I - X(X'X)inverseX' to clarify: '=transpose; inverse of quantity (X'X)"

Sorry but it's incomprehensible.
 

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