#linear algebra

Question

why Eigen Vectors can not be zero ?

Answer 1

$0$ vector is called a trivial  eigen vector and it exists for any matrix . But when we talk about eigen vectors, in general, it always refers to non zero/ non trivial eigen vectors

Answer 2

Let's first Define Eigen vector, which is.

An eigenvector is a non-zero vector within a vector space that, when multiplied by a given square matrix, results in a scaled version of itself. 
Av=λv

Let's just remove the 'non-zero vector' part for a while and lets feel practically, what is an eigenvector.

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So, eigenvector is a direction within a space, that remains unchanged when any linear transformation is applied.
Again, eigenvector is a direction.

Can direction be 0?
Can direction be Null?
Can direction be empty?

I'll ask a more simpler real life question.

We are living in a 3-D world.
1. Close one eye.
2. Look constant in one direction of your choice.
3. Draw a (x,y,z) graph in your mind with (0,0,0) where your open eye is.
Can you say that you are looking at (0,0,0). You can't because you cannot see your open eye from your open eye. IMPOSSIBLE. 

Now, I'll give another practical example.

Imagine you have a square piece of paper with an arrow drawn on it. Now, you apply some transformation to this paper, like folding, stretching (let's say its a stretchable paper), or rotating. An eigenvector in this case would be a direction on the paper where the arrow remains pointing in the same direction after the transformation, possibly getting longer or shorter.

1. Stretching: If you stretch the paper horizontally, the arrow might get longer or shorter, but there might be a direction on the paper where the arrow stays pointing in the same direction. That direction is an eigenvector.

2. Rotation: If you rotate the paper, there might be a direction where the arrow remains pointing in the same direction relative to the paper's new orientation. That direction is another eigenvector.

3. Folding: Even if you fold the paper, there might be a crease along which the arrow remains aligned. That crease direction is also an eigenvector.

Now, read this line again
The eigenvector represents a direction on the paper that remains relatively unchanged after the transformation, except for a possible scaling (stretching or compressing), which is represented by the eigenvalue

Ofcourse, this was to make anyone feel what an eigenvector and eigenvalue is. The real answer to this question is that the poor guy is defined to be a non-zero vector.

 

Answer 3

A matrix (A-lambdai)x=0 can have only solution as 0 ( trivial solution ) if (a - lambdai) has determinant non zero ( the column vectors of a - lambdai are linearly independent ) then there will be no eigen values as we can never find any characteristic eqn and find values of lambda .

Answer 4

If we say about the eigen vector, it is a type of vector that when multiplied by the other matrix, will either increase or decrease the length of the original matrix, even may change its direction..., being a unique vector, remaining unchanged, when any LT is applied.. Hence, in other words, it shows the direction of increament or decrement of the original matrix. And as we know that the direction of the line(representing the matrix) can't be zero, so, it is never considered as 0.
Above all, considering the uniqueness, for (A-(lamda*I))*x = 0, where x is Eigen Vector, we can say that X can take value 0 for each and every matrix A, and the uniqueness of the matrix is dissolved out, and hence, the matrix though may be represented by any line in the plane, will always be held upto (0,0) if multiplied by X=0..