Decompose Involutory Matrix into Difference of Two Idempotents

In summary, the conversation discusses proving that an involutory matrix can be expressed as the difference of two idempotent matrices. The given equation to decompose the involutory matrix is used to show that each of the resulting matrices is idempotent.
  • #1
christang_1023
27
3
Homework Statement
A square matrix ##A## is said to be
• an involutory matrix if ##A^2 = I##,
• an idempotent if ##A^2 = A##.
Show that every involutory matrix can be expressed as a difference of two idempotents.
Relevant Equations
The involutory matrix can be decomposed to ##A=\frac{1}{2}(I+A)-\frac{1}{2}(I-A)##
I feel confused about proving the two terms are idempotents.
 
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  • #2
christang_1023 said:
Homework Statement: A square matrix ##A## is said to be
• an involutory matrix if ##A^2 = I##,
• an idempotent if ##A^2 = A##.
Show that every involutory matrix can be expressed as a difference of two idempotents.
Homework Equations: The involutory matrix can be decomposed to ##A=\frac{1}{2}(I+A)-\frac{1}{2}(I-A)##

I feel confused about proving the two terms are idempotents.
You're given an involutory matrix A, which means that ##A^2 = I##. Clearly A can be decomposed into ##\frac{1}{2}(I+A)-\frac{1}{2}(I-A)##.

Look at each of the matrices ##\frac{1}{2}(I+A)## and ##\frac{1}{2}(I-A)## and show that each of them is an idempotent matrix.
 

Related to Decompose Involutory Matrix into Difference of Two Idempotents

1. What is a decomposed involutory matrix?

A decomposed involutory matrix is a square matrix that can be expressed as the difference of two idempotent matrices. This means that when the matrix is multiplied by itself, the result is equal to the original matrix, making it an involutory matrix.

2. How is a decomposed involutory matrix useful?

A decomposed involutory matrix can be useful in various mathematical and scientific applications, such as in linear algebra, cryptography, and signal processing. It allows for easier manipulation and analysis of the original matrix, as well as providing insight into its properties.

3. How do you decompose an involutory matrix into a difference of two idempotents?

To decompose an involutory matrix into a difference of two idempotents, you can use the spectral theorem. This involves finding the eigenvalues and eigenvectors of the matrix, and then using them to construct the idempotent matrices.

4. Are there any limitations to decomposing an involutory matrix?

Yes, there are limitations to decomposing an involutory matrix. One limitation is that not all involutory matrices can be decomposed into a difference of two idempotents. Additionally, the decomposition may not be unique, meaning that there can be multiple ways to express the matrix as a difference of two idempotents.

5. Can a decomposed involutory matrix be reconstructed into the original matrix?

Yes, a decomposed involutory matrix can be reconstructed into the original matrix by simply adding the two idempotent matrices. This is because the decomposition is essentially a way of expressing the original matrix in a different form, but the information and properties of the original matrix are still preserved.

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