Showing that S is an Eigenvalue of a Matrix

  • #1
Drakkith
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Homework Statement


Consider an n x n matrix A with the property that the row sums all equal the same number S. Show that S is an eigenvalue of A. [Hint: Find an eigenvector.]

Homework Equations


##Ax=λx##

The Attempt at a Solution


S is just lambda here, so I begin solving this just like you would normally.
##Ax=Sx##
##Ax-Sx = 0##
##(A-SI)x = 0##

Subtracting gives me the matrix: ##\begin{bmatrix}
a_{11}-S & a_{12} & a_{13} \\
a_{21} & a_{22}-S & a_{23} \\
a_{31} & a_{32} & a_{33}-S
\end{bmatrix}##
My problem is that I don't know how to find an eigenvector from this matrix. I can't row reduce because I don't know any of the values of the matrix, and I can't recall any other way to find the eigenvector.
 
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  • #2
This one is tricky, but almost simple. I took a lucky guess after looking at it for about 5 minutes and got lucky. One hint: Try a simple eigenvector, but not one that has mostly zeros=in fact, try a very simple eigenvector without any zeros... I may give you an additional hint if you don't see what the eigenvector is that works...
 
  • #3
Charles Link said:
This one is tricky, but almost simple. I took a lucky guess after looking at it for about 5 minutes and got lucky. One hint: Try a simple eigenvector, but not one that has mostly zeros=in fact, try a very simple eigenvector without any zeros... I may give you an additional hint if you don't see what the eigenvector is that works...

Hmm. Well, using a matrix of all 1's, I get S=3 and the eigenvector is ##x_3
\begin{bmatrix}
1\\
1\\
1
\end{bmatrix}##
 
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  • #4
Drakkith said:
Hmm. Well, using a matrix of all 1's, I get S=3 and the eigenvector is ##x_3
\begin{bmatrix}
1\\
1\\
1
\end{bmatrix}##
## S ## will not be equal to 3 and/or n. ## S ## will be equal to what each row sums to. ## S ## is the eigenvalue. And yes, you found the correct eigenvector ! :)
 
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  • #5
Drakkith said:

Homework Statement


Consider an n x n matrix A with the property that the row sums all equal the same number S. Show that S is an eigenvalue of A. [Hint: Find an eigenvector.]

Homework Equations


##Ax=λx##

The Attempt at a Solution


S is just lambda here, so I begin solving this just like you would normally.
##Ax=Sx##
##Ax-Sx = 0##
##(A-SI)x = 0##

Subtracting gives me the matrix: ##\begin{bmatrix}
a_{11}-S & a_{12} & a_{13} \\
a_{21} & a_{22}-S & a_{23} \\
a_{31} & a_{32} & a_{33}-S
\end{bmatrix}##
My problem is that I don't know how to find an eigenvector from this matrix. I can't row reduce because I don't know any of the values of the matrix, and I can't recall any other way to find the eigenvector.
You certainly know that the equation ##(A-SI)x = 0## has nontrivial solutions if the determinant ##|A-SI| = 0##
Remember what operations do not change the value of a determinant. Read the problem. "Consider an n x n matrix A with the property that the row sums all equal the same number S."
 

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