Find eingenvalues and eigenvectors of an order n matrix

In summary: The eigenvectors are those perpendicular to (1, 1, ..., 1). That is, x1+ x2+ ...+ xn= 0. Of course, that is an n-1 dimensional space and we can choose any n-1 independent vectors from it as a basis. One obvious choice is to let xi= 1, xj= -1, xk= 0 for i, j, k distinct and the rest 0. There are n choose 2 ways to do that for n= 4, (1, 0, 0, -1), (0, 1, 0, -1), and (0, 0, 1
  • #1
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Homework Statement



Being [itex]T\in L(\mathbb{R}^n)[/itex] a linear operador defined by [itex]T(x_1, ... ,x_n )=(x_1+...+x_n,...,x_1+...+x_n )[/itex], find all eigenvalues and eigenvectors of T.

Homework Equations



[itex]det(T-\lambda I)=0, Ax=\lambda x[/itex]

The Attempt at a Solution



By checking n=1,2,3,4 I guess the answer is:

λ=n, x=(1,1,1)
λ=0 (multiplicity n-1), x such as , [itex]\forall k \in \{1,...,(n-1)\}[/itex], [itex]x_k=1[/itex], [itex]x_n=-1[/itex] and [itex]x_i=0[/itex] in all other positions. For instance, for n=4, we have (1,0,0,-1), (0,1,0,-1), (0,0,1,-1).

But how do I prove it for the general case? I'm trying induction, but I think I'm missing something...

Thanks in advance! :)
 
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  • #2
This operator replaces every vector with the vector having each component the sum of the original vectors components. The crucial point is that the image of the vector space under this linear transformation is the space of all vectors of the form (x, x, x, ..., x) with all components the same. That is one dimensional and has {(1, 1, 1, ..., 1)} as a basis. In particular, it maps (1, 1, 1, ..., 1) into the vector (n, n, ..., n) so, as you say, one eigenvalue is n and a corresponding eigenvector is (1, 1, ..., 1).
Since this maps all of Rn into a single line, there is a second, obvious eigenvalue with n-1 dimensional "eigenspace".
 
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Related to Find eingenvalues and eigenvectors of an order n matrix

What is the difference between eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are both related to the properties of a matrix, but they represent different aspects. Eigenvalues are scalars that represent how a matrix stretches or compresses a vector. Eigenvectors are the corresponding vectors that do not change direction when multiplied by the matrix.

What is the significance of finding eigenvalues and eigenvectors of a matrix?

Finding eigenvalues and eigenvectors can help in understanding the behavior and properties of a matrix. They are used in various fields such as physics, engineering, and computer science for solving systems of linear equations, identifying patterns, and analyzing data.

How do you find eigenvalues and eigenvectors of a matrix?

To find the eigenvalues of a matrix, you need to solve the characteristic equation, which involves finding the determinant of the matrix. Once the eigenvalues are known, the corresponding eigenvectors can be found by solving a system of linear equations using the eigenvalues as coefficients.

What is the relationship between eigenvalues and eigenvectors of a matrix?

The eigenvalues and eigenvectors of a matrix are closely related. Each eigenvalue has a corresponding eigenvector, and they form an eigenpair. The eigenvectors are the basis for the eigenspace, which is the set of all vectors that are only scaled by the eigenvalue when multiplied by the matrix.

Can all matrices be diagonalized to find eigenvalues and eigenvectors?

No, not all matrices can be diagonalized to find eigenvalues and eigenvectors. The condition for diagonalizability is that the matrix must have n linearly independent eigenvectors, where n is the order of the matrix. If this condition is not met, alternative methods must be used to find the eigenvalues and eigenvectors.

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