Guess Eigenvalue of 2x2 Matrix with Constant k Sum

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In summary, A is a 2x2 matrix with a constant k such that the sum of the entries in each row and column is k. The given vectors a, b, and c may or may not be eigenvectors of A depending on their effects on the matrix. The corresponding eigenvalue can be determined after determining the eigenvector.
  • #1
fk378
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Homework Statement


Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvector of A?

a. [1,0]
b. [0,1]
c. [1,1]

(The answer can be any or all of these)

The Attempt at a Solution


I don't know how to approach this...
 
Last edited:
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  • #2
fk378 said:
Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvalue of A?

a. [1,0]
b. [0,1]
c. [1,1]

(The answer can be any or all of these)
…
I don't know how to approach this...

Hi fk378! :smile:

Sorry to say this, but … just go for the screamingly obvious :rolleyes::

for a 2x2 matrix ab cd, what is its effect on each of the three given vectors? :wink:
 
  • #3
fk378 said:


a. [1,0]
b. [0,1]
c. [1,1]

...


...and here i thought that eigenvalues were scalars
 
  • #4
Yes, it is. However, fk378 did ask "Which of the following must be an eigenvector of A?
Presumably, after determining which of those vectors is an eigevector, you could then determine the corresponding eigenvalue. That's certainly better than "guessing" the eigenvalue!

fk378, what is
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}[/tex]?
What is
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}0 \\ 1\end{bmatrix}[/tex]?
What is
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 \\ 1\end{bmatrix}[/tex]?

Remembering, of course that a+ c= k, b+ d= k, a+ b= k, and c+ d= k.
 
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  • #5
AH sorry! I had a typo in the subject line...it should read "guess the eigenvector"

So sorry for the confusion here.
 
  • #6
HallsofIvy said:
Yes, it is. However, fk378 did ask "Which of the following must be an eigenvector of A?
.


Well, the OP defenitely edited it at some latter point, since when i looked it it read eigenvalue.
 
  • #7
I wasn't confused :blushing:

anyway, what's your answer …

for a 2x2 matrix ab cd, what is its effect on each of the three given vectors? :wink:
 

FAQ: Guess Eigenvalue of 2x2 Matrix with Constant k Sum

What is an eigenvalue?

An eigenvalue is a scalar value that represents the scale factor by which a vector is stretched or compressed when multiplied by a matrix. In other words, it is a value that defines the behavior of a matrix when applied to a vector.

How do you find the eigenvalues of a 2x2 matrix?

To find the eigenvalues of a 2x2 matrix, you can use the characteristic equation, which is given by:
λ2 - (a+d)λ + (ad-bc) = 0
where a, b, c, and d are the elements of the matrix. Solve this equation to find the two eigenvalues.

What is the significance of the eigenvalues in a 2x2 matrix?

The eigenvalues of a 2x2 matrix provide important information about the behavior of the matrix. They can tell us about the stretching or compression of vectors when multiplied by the matrix, the direction of the transformation, and whether the matrix is invertible or not.

How does the constant k affect the eigenvalues of a 2x2 matrix?

The constant k in a 2x2 matrix with the form [a, b; c, d] + kI, where I is the identity matrix, affects the eigenvalues by shifting them by a factor of k. This means that the eigenvalues of the original matrix [a, b; c, d] will be equal to the eigenvalues of [a+k, b; c, d+k] + kI.

Can the eigenvalues of a 2x2 matrix be complex numbers?

Yes, the eigenvalues of a 2x2 matrix can be complex numbers. In fact, if the matrix has complex elements, it is likely that the eigenvalues will also be complex. This is because complex numbers allow for rotation and scaling in two dimensions, which is the behavior that eigenvalues represent.

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