Very interesting. So the eigenspace that corresponds to the eigenvalue 3 is a plane in R3. It would be the same, it'd be 3 times this length, but in the opposite direction. So let's put this in reduced row echelon form. So minus 2 times 2 plus 1 is 0.
How to find eigenvalues given eigenvectors
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen- is applied liberally when naming them: The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation. Let me write this way. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix , for example by diagonalizing it. So minus 2 minus minus 2 is 0. So let me move this lower down where I have some free real estate. So we had that 3 by 3 matrix, A. And then we have minus 2 minus minus 5. This would be true for any of these guys. Actually let me just do it this way. So if you apply the matrix transformation to any of these vectors, you're just going to scale them up by 3.
And so these two rows, or these two equations, give us no information. This must be true but this is easier to work with.
Eigenvalues and eigenvectors pdf
So we had that 3 by 3 matrix, A. So v2 is equal to a times 1. It's a line like that. And then the last row is 0, 0, 0. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today. This guy's null space is going to be the null space of that guy in reduced row echelon form. So we get 1, 1, 1. So the null space of this matrix is the eigenspace. Minus 3 minus 2 is minus 5. So v1, v2, v3 are going to be equal to the 0 vector. It represents some transformation in R3.
This matrix right here-- I've just copied and pasted from above. And then let me do the last row in a different color for fun. Just to be a little bit formal about it.
Let's say that this is x right there.
Which is not this matrix. So what is-- the null space of this is the same thing as the null space of this in reduced row echelon form So let's put it in reduced row echelon form.
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