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arthurhenry
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Why is the Trace of a projection is its Rank.
Thank you
Thank you
Why Rank is the Trace of a Projection
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SUMMARY
The rank of a projection operator P is defined as the number of its eigenvalues that equal 1, due to the property that P satisfies P² = P. In the Jordan normal form of P, the trace, which is the sum of all eigenvalues, directly corresponds to the rank of P. This establishes a definitive relationship between the trace and the rank of projection operators in linear algebra.
PREREQUISITES- Understanding of projection operators in linear algebra
- Familiarity with eigenvalues and eigenvectors
- Knowledge of Jordan normal form
- Basic concepts of linear transformations
- Study the properties of projection operators in linear algebra
- Learn about the Jordan normal form and its applications
- Explore the relationship between trace and eigenvalues in matrices
- Read "Algebras of Linear Transformations" by Douglas Farenick for deeper insights
Students and professionals in mathematics, particularly those focusing on linear algebra, as well as educators seeking to clarify concepts related to projection operators and their properties.
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Hi arthurhenry! 
A projection operator P satisfies P2=P. So it's only eigenvalues are 0 and 1. It is easy to see that the rank of P is the number of eigenvalues that are 1. Thus the sum of the eigenvalues is in this case the rank...
Now, take the Jordan normal form of P, then the diagonal contains all the eigenvalues. In particular, the trace is the sum of all the eigenvalues. And thus equals the rank of P.
A projection operator P satisfies P2=P. So it's only eigenvalues are 0 and 1. It is easy to see that the rank of P is the number of eigenvalues that are 1. Thus the sum of the eigenvalues is in this case the rank...
Now, take the Jordan normal form of P, then the diagonal contains all the eigenvalues. In particular, the trace is the sum of all the eigenvalues. And thus equals the rank of P.
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arthurhenry
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Dear Micromass,
I thank you for your help --on two occasions now, as you answered another post of mine. Some of these questions come as I verify a comment or at times directly a trying to do an exercise. I am reading a book "Algebras of Linear Transformations" by Douglas Farenick, to teach myself some of that material. I do realize some of my questions are rather rudimentary, I apologize.
I thank you for your help --on two occasions now, as you answered another post of mine. Some of these questions come as I verify a comment or at times directly a trying to do an exercise. I am reading a book "Algebras of Linear Transformations" by Douglas Farenick, to teach myself some of that material. I do realize some of my questions are rather rudimentary, I apologize.
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Don't apologize! It's only by asking such a questions that you'll learn the material. Everybody has to go through it 
It's only by asking such a questions that you'll learn the material. Everybody has to go through it Similar threads
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