Linear Algebra- Diagonal Matrix

In summary, the conversation discusses how to prove that a diagonal matrix, with diagonal entries of either 0 or 1, is idempotent. It also shows that if a nonsingular matrix is multiplied by such a diagonal matrix and its inverse, the resulting matrix is also idempotent. There is also a discussion about the notation for matrix multiplication.
  • #1
Roni1985
201
0

Homework Statement



Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1

a) Show that D is idempotent

b) Show that if X is a nonsingular matrix and A=XD(X)-1 , then A is idempotent

Homework Equations


The Attempt at a Solution



a) I tried it, and it works for a specific matrix, say 3x3... but I don't know if it's really a proof.
I need to show that

D2=D

b) I solved this part

Also,
is (AB)2= A2*B2
or
(AB)2=B2*A2
?
 
Last edited:
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  • #2
You just need to make up some notation. Let diag(a1,a2,...an) be the matrix whose diagonal entries are a1, a2... Now what diag(a1,a2,...an)*diag(b1,b2,...bn)? (AB)^2 generally isn't equal to either A^2*B^2 or B^2*A^2. It's just ABAB.
 
  • #3
Dick said:
You just need to make up some notation. Let diag(a1,a2,...an) be the matrix whose diagonal entries are a1, a2... Now what diag(a1,a2,...an)*diag(b1,b2,...bn)? (AB)^2 generally isn't equal to either A^2*B^2 or B^2*A^2. It's just ABAB.

oh, right , it works ...
thanks a lot for the help ...
 

1. What is a diagonal matrix?

A diagonal matrix is a type of square matrix where all the elements outside of the main diagonal (running from the top left to the bottom right) are equal to zero. The main diagonal elements can be any real numbers.

2. What is the significance of diagonal matrices in linear algebra?

Diagonal matrices are important in linear algebra because they simplify many calculations and operations. They are particularly useful in solving systems of linear equations, finding eigenvalues and eigenvectors, and in diagonalization of matrices.

3. How do you determine if a matrix is diagonal or not?

A matrix is diagonal if and only if all the elements outside of the main diagonal are equal to zero. This can be checked by looking at the entries in the matrix or by calculating the determinant. If the determinant is equal to zero, then the matrix is diagonal.

4. Can a non-square matrix be diagonal?

No, a non-square matrix cannot be diagonal. Diagonal matrices are only defined for square matrices, meaning that the number of rows and columns must be equal.

5. How are diagonal matrices used in real-world applications?

Diagonal matrices have many applications in fields such as physics, engineering, economics, and computer science. They are used to solve systems of linear equations in mathematical modeling, in signal processing and image compression, and in quantum mechanics for representing observables and transformations.

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