Exploring the Relationship Between Trace and Rank in Idempotent Matrices

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In summary, for an idempotent matrix A, trace(A) = rank(A) and A is diagonalizable with eigenvalues of either 0 or 1. To learn more about these relationships, one can refer to a book on matrix analysis or inference. These concepts may not have been covered in a linear algebra course, but may be mentioned in a course on regression analysis.
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maverick280857
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Hi

Is it true that for an idempotent matrix [itex]A[/itex] (satisfying [itex]A^2 = A[/itex]), we have

[tex]trace(A) = rank(A)[/tex]

Where can I find more general identities or rather, relationships between trace and rank? I did not encounter such things in my linear algebra course. I'm taking a course on regression analysis this semester and that's where I ran into it.

I'd appreciate if someone could point me to a book on matrix analysis or inference where these things would be mentioned in some detail. For some reason, the more "practically relevant" results were not covered in my freshman math courses.

Thanks in advance.
Cheers
Vivek.
 
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Hint 1: an idempotent matrix is diagonalizable.
Hint 2: the eigenvalues of an idempotent matrix are either 0 or 1.
 
  • #3
radou said:
Hint 1: an idempotent matrix is diagonalizable.
Hint 2: the eigenvalues of an idempotent matrix are either 0 or 1.

Thanks..yes, I thought of the identity matrix and it all made sense.
 

1. What is the definition of rank of an idempotent matrix?

The rank of an idempotent matrix is defined as the number of linearly independent rows (or columns) in the matrix. In other words, it is the maximum number of rows (or columns) that can be chosen from the matrix such that they are linearly independent.

2. How is the rank of an idempotent matrix related to its eigenvalues?

The rank of an idempotent matrix is equal to the number of eigenvalues that are equal to 1. This is because an idempotent matrix is one that can be written as A^2 = A, which means that its eigenvalues must satisfy the equation λ^2 = λ. Therefore, the only possible eigenvalues are 0 and 1, and the rank is equal to the number of eigenvalues equal to 1.

3. Can an idempotent matrix have a rank of 0?

No, an idempotent matrix cannot have a rank of 0. This is because the definition of an idempotent matrix requires it to have at least one eigenvalue equal to 1, which would correspond to at least one linearly independent row (or column) and a minimum rank of 1.

4. How is the rank of an idempotent matrix related to its trace?

The rank of an idempotent matrix is equal to its trace (the sum of its diagonal elements). This is because the trace of an idempotent matrix is equal to the number of eigenvalues equal to 1, which, as mentioned in question 2, is equal to the rank of the matrix.

5. Is the rank of an idempotent matrix affected by elementary row (or column) operations?

No, the rank of an idempotent matrix is not affected by elementary row (or column) operations. This is because these operations do not change the linear independence of the rows (or columns) of the matrix, and therefore, do not change the rank.

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