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There is a similiar post eigen values and eigen vectors in case of matrixes and differential equations, but there isnt any answer.

I am trying to understand the concept of eigen value and eigen vectors. The word eigen means own, but how can you explain an "own" value for an equation? Are these the solution of an equation? If yes, then why not simply use the term solution or roots?

Also, the visualisation part is difficult. For example, the roots of the quadratic equation are the parts where the graph intesects the x axis ( y = 0 ) . Is there any way, one can visualise eigen values? Or they are completely different?

Some terminology confuses me further. For example, quad means 4, and there is no fourth order in quadratic equation. So, why isnt quadratic equation not called duotic equation. It would make more sense to me.

Further reading materials would be helpful, that are easy to understand for beginners.

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Suppose we have a linear transform from $\mathbb R^n\to\mathbb R^n$ defined by a matrix $A$. This transform maps each vector $\vec v\in\mathbb R^n$ to a new vector $A\vec v\in\mathbb R^n$.

An eigenvector of $A$ is simply a nonzero vector $\vec v$ such that $\vec v$ and $A\vec v$ are parallel. Since two parallel vectors are scalar multiples of each other, we can write this statement as an equation, called the characteristic equation of $A$.

$$A\vec v=\lambda\vec v$$

we can rearrange this equation using the fact that $\vec v=I\vec v$, where $I$ is the Identity matrix.

$$(A-\lambda I)\vec v=0$$

Where $\vec v$ is a nonzero vector and $\lambda$ is a constant. This equation will typically have several solutions for $\vec v$ and $\lambda$, which we'll call the eigenvectors $\{\vec v_1,\vec v_2,...\}$ and eigenvalues $\{\lambda_1,\lambda_2,...\}$. Each eigenvector has exactly one corresponding eigenvalue, but one eigenvalue can correspond to many eigenvectors.

If $\vec v_1$ is an eigenvector with eigenvalue $\lambda_1$, we can see from the equation above that $c\vec v_1$ will also be an eigenvector with the same eigenvalue. If $\vec v_2$ is an eigenvector with the same eigenvalue, then any linear combination of $a\vec v_1+b\vec v_2$ is also an eigenvector with the same eigenvalue. Because of this, we almost always restrict ourselves to a linearly independent set of eigenvectors. One matix can have many sets of eigenvectors, but the eigenvalues will always be the same. The span of all the eigenvectors corresponding to a particular eigenvector will always be the same for a given matrix.

For certain special cases, it is necessary to consider generalized eigenvectors in order to completely describe the transform. These obey the characteristic equation $(A-\lambda I)^n\vec v=0$ for some $n\in\mathbb N$.

According to Wikipedia, the term "eigen-" is German for "characteristic", among other things. The choice makes sense, in that no only do the eigenvalues and eigenvectors completely describe a linear transform, but they also characterize what the transformation does geometrically. For example, real eigenvalues correspond to scaling factors, and the eigenvectors tell us in which direction this scaling is applied. Imaginary eigenvalues correspond to rotations, and the eigenvectors tell us the plane of rotation.

Of course, this concept generalizes to linear transforms that are harder to visualize. Of particular importance to differential equations are linear differential operators which are linear transformations on function spaces. The notion of eigenvectors and eigenvalues are still quite useful there, they just have to be adapted to infinite dimensions.

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  • $\begingroup$ There do exist linear operators that can not be diagonalized. Those are not completely described by the eigenvalues. But other than that you have a very good description. $\endgroup$ – mathreadler Aug 28 '17 at 7:27
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An eigenvector to a matrix is a vector that when multiplied with the matrix maps onto itself up to scaling with a scalar of the underlying field.

Eigen is the german word for own. It basically means something that maps back to itself.

What you need to learn to understand this language in the context of differential equations is the connection between functions and vectors and differentiation and linear operators. In my experience it takes time for it to grow in the mind so don't panick if you don't understand it right away.

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  • $\begingroup$ Why do you use the preposition "to" in "an eigenvector to a matrix" ? Can you please give a very small example? Would you please be kind to write the answer in simpler english? Maybe breaking long sentences into short ones. Would be grateful. $\endgroup$ – infoclogged Aug 26 '17 at 12:52
  • $\begingroup$ English is not my native language so I don't know how much it would help. $\endgroup$ – mathreadler Aug 26 '17 at 14:38
  • $\begingroup$ same here. but maybe someone else can do it for us. thanks for your time anyways. $\endgroup$ – infoclogged Aug 26 '17 at 14:44
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To answer the last part about quadratic equation:

An equation about a quadrat, kvadrat : in many languages other than english it is a geometric shape: a rectangle - which has four sides and four corners. Iff we assume linear expressions as it's side and asking for it's area, becomes a quadratic equation:

$$(ax+b)(cx+d) = A \Leftrightarrow \\acx^2+(ad+bc)x+bd = A$$

Which is a quadratic equation.

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