Showing that S is an Eigenvalue of a Matrix

[Drakkith]

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.

 

[Charles Link]

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...

 

[Drakkith]

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 $x_3 \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}$

 

[Charles Link]

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 ! :)

 

[ehild]

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."