Follow up to B=I-2A-A^2 where mu=1-2lambda+(lambda)^2

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The determinant of B-\mu I is equal to 0, which means that B-\mu I is not invertible. Since \lambda=1, \mu=0, this shows that B is singular.
  • #1
Dustinsfl
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Show that if [tex]\lambda=1[/tex] is an eigenvalue of A, then the matrix B will be singular.

[tex]\mu(1)=1-2+1=0[/tex]

[tex]B\mathbf{x}=0\mathbf{x}[/tex]

or does this need to be done via determinant?

[tex]det(B-\mu I)=0[/tex] since [tex]\lambda=1, \mu=0[/tex].

[tex]det(B-0I)=det(B)=0[/tex] Hence B is singular.
 
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  • #2
Dustinsfl said:
Show that if [tex]\lambda=1[/tex] is an eigenvalue of A, then the matrix B will be singular.

[tex]\mu(1)=1-2+1=0[/tex]

[tex]B\mathbf{x}=0\mathbf{x}[/tex]
This is true, but how did you arrive at it? Also, what's the significance of this equation in terms of what you're supposed to show?
Dustinsfl said:
or does this need to be done via determinant?

[tex]det(B-\mu I)=0[/tex] since [tex]\lambda=1, \mu=0[/tex].

[tex]det(B-0I)=det(B)=0[/tex] Hence B is singular.
This is another way to show it.
 

What is the formula for "Follow up to B=I-2A-A^2"?

The formula is B=I-2A-A^2, where B represents a matrix, I is the identity matrix, and A is a matrix with dimensions compatible with B.

What is the significance of mu in this formula?

Mu represents the eigenvalue of the matrix B. It is a parameter that affects the behavior of the matrix and can help determine its properties, such as stability and convergence.

What role does lambda play in this formula?

Lambda represents the eigenvalue of the matrix A. It is also a parameter that affects the behavior of B and can help determine its properties.

Can this formula be used to solve for the eigenvalues of B?

Yes, this formula can be used to solve for the eigenvalues of B. The eigenvalues can be found by setting B equal to 0 and solving for lambda using the quadratic formula.

Are there any specific conditions or assumptions that need to be met for this formula to be valid?

Yes, in order for this formula to be valid, the matrices B and A must be square matrices with compatible dimensions. Additionally, the eigenvalues of B must be distinct and the matrix A must be diagonalizable.

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