What is the Determinant of an Idempotent Matrix?

In summary, the conversation discusses the concept of idempotent matrices and how to show that det(P)=0 if P is idempotent and not equal to the identity matrix. It involves using the fact that if det(P) is not equal to 0, then P is invertible. The conversation also suggests using the matrix A=I-X(X'X)^-1X' to determine if it is idempotent.
  • #1
forty
135
0
A matrix P is called idempotent if P^2 = P. If P is idempotent and P =/= I show that det(P)=0.

I don't really know where to go with this but i have a feeling that it involves taking the det of each side.

det(P^2) = det(P)
det(P)det(P) = det(P)

where to from here if that's even the right step/method to take, or if its even right at all >_>

Thanks :)
 
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  • #2
looks fine to me; now for what values of det(P) does that equation hold?
 
  • #3
det(P) = 1 or 0
 
  • #4
Okay, if detP=1, and P^2=P, what matrix must P be?
 
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  • #5
Hint: use the fact that if [itex]det(P) \neq 0[/itex], then P is invertible. Multiply [itex]P^2=P[/itex] by [itex]P^{-1}[/itex].
 
  • #6
but it says det(P)=/=1. How do you show that det(P)=0?
 
  • #7
det(P2) = det(P)

=> det(P)^2-det(P)=0

This is the same as t^2-t=0 where t=det(P). Factorise and use that fact that P=/= I
 
  • #8
I have a questions:
if A=I-X(X'X)^-1X'
is it A idempotent?
 
  • #9
kendarto: don't jump into another poster's thread.

try to calculate [tex] A^2 [/tex] and answer this for yourself.
 

1. What is an idempotent matrix?

An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix. In other words, the matrix remains unchanged after the multiplication.

2. How do you prove that a matrix is idempotent?

To prove that a matrix is idempotent, you need to multiply the matrix by itself and show that the resulting matrix is the same as the original matrix. This can be done by performing the matrix multiplication operation and comparing the resulting matrix with the original matrix.

3. What is the significance of an idempotent matrix?

Idempotent matrices have several important applications in linear algebra and data analysis. They are particularly useful in projection operations and in representing systems that exhibit self-similarity or self-replication.

4. Can a non-square matrix be idempotent?

No, a non-square matrix cannot be idempotent. Only square matrices (with an equal number of rows and columns) can be idempotent.

5. How can we use the concept of idempotent matrices in real-world problems?

Idempotent matrices are commonly used in data analysis and machine learning algorithms, particularly in dimensionality reduction techniques such as principal component analysis. They can also be used in optimization problems and in representing the steady state of a system in physics or engineering.

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