Solve Idempotent Matrices: 3x3 X1, X2, X3 and Show AXi=Xi

  • Thread starter sam.baranoff
  • Start date
  • Tags
    Matrices
In summary, the conversation is about finding the value of A when multiplied by different idempotent matrices X1, X2, and X3. The question is to show that AXi = Xi and the participants are discussing their solutions, with one pointing out a potential mistake in the statement of the problem.
  • #1
sam.baranoff
2
0
Guys, I really need your help. I've been working on this problem all day, and I'm starting to pull my hair out.

I have three idempotent 3x3 matrices X1, X2 and X3.
X1=[1,1,1;0,0,0;0,0,0]
X2=[0,0,0;1,1,1;0,0,0]
X3=[0,0,0;0,0,0;1,1,1]

Let A=aX1+bX2+cX3 where a, b, c are scalars

Show that AXi=Xi (where i=1, 2, 3)

I get A=[a,a,a;b,b,b;c,c,c], but then I have AXi=A... What am I doing wrong?
 
Physics news on Phys.org
  • #2
welcome to pf!

hi sam! welcome to pf! :smile:
sam.baranoff said:
I have three idempotent 3x3 matrices X1, X2 and X3.
X1=[1,1,1;0,0,0;0,0,0]
X2=[0,0,0;1,1,1;0,0,0]
X3=[0,0,0;0,0,0;1,1,1]

Let A=aX1+bX2+cX3 where a, b, c are scalars

Show that AXi=Xi (where i=1, 2, 3)

I get A=[a,a,a;b,b,b;c,c,c], but then I have AXi=A... What am I doing wrong?

yes, i get the same :smile:

it must be a misprint for "Show that AXi = A" :rolleyes:
 

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, A² = A.

What is the significance of idempotent matrices?

Idempotent matrices are important in linear algebra and matrix theory because they have many useful properties and applications. They are commonly used in statistics, optimization, and computer graphics.

How do you solve for idempotent matrices?

To solve for idempotent matrices, we can use the following steps:

  • Write out the given matrix in the form of A = [aij], where i and j represent the rows and columns, respectively.
  • Multiply the matrix by itself, A² = AA = [aijajk].
  • Set A² = A and solve for the unknown variables.
  • Verify that the resulting matrix satisfies the idempotent property, A² = A.

How many solutions are possible for idempotent matrices?

There are infinitely many solutions for idempotent matrices, as any matrix with 0s and 1s along the main diagonal will satisfy the idempotent property. However, there may also be other solutions depending on the specific values of the matrix elements.

Can idempotent matrices have different dimensions?

No, idempotent matrices must be square matrices, meaning they have the same number of rows and columns. This is because the idempotent property can only be satisfied when the matrix is multiplied by itself, which can only be done with square matrices.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Replies
9
Views
1K
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
2
Views
4K
Back
Top