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# Finding eigenvectors and eigenspaces example

Finding the eigenvectors and eigenspaces of a 2x2 matrix. Created by Sal Khan.

## Want to join the conversation?

• What is a null space?
• A null space is commonly referred to as the "kernel" of some function and is the set of all points that send a function to zero i.e all x's such that f(x)= 0 is true.

In terms of linear algebra/linear transformation/matrix algebra you can think of a null space (or kernel) as the set of all vectors v such that

Av=0 where A is a mxn matrix and 0 is the zero vector.
• at , why does the Eigenvector equal span (1/2, 1), not span (1, - 1/2)?
• these are equivalent, since (1,-1/2) is in the span of (1/2,1) and vice versa. So it doesn't matter which one you choose, both statements are correct.
• I understand most of the process, but I feel like the initial question was unanswered. Could someone help me?

So, if I understand correctly, our initial question was "What are the Eigenvalues (lambda) and Eigenvectors (v) that satisfy the equation T(v) = A*v = lambda*v?"

We found the values of lambda that are possible in the previous video (link at bottom).

We then used each distinct possible value of lambda, and plugged it back in to the equation [A-(lambda*I)]v = 0 to determine all possible vectors v that would make that work (the null space).

This is where I get confused. Sal ends up talking in the end of the video (starting about ) about the span of these vectors, or the Eigenspace. He acts as if the Eigenspace IS the answer.

Well, is it?

The initial question was "What are the Eigenvalues (lambda) and Eigenvectors (v) that satisfy the equation T(v) = A*v = lambda*v?"

I think that the Eigenspaces would accommodate all combinations of possible Eigenvalues and Eigenvectors, but am I wrong in assuming that? Would we have to specify what the Eigenvalues are? I feel comfortable listing a span as an answer to the set of all possible Eigenvectors, but I feel like I'm not accounting for the 2 distinct Eigenvalues.

Or am I just wrong in what the initial question was?

• one point of finding eigenvectors is to find a matrix "similar" to the original that can be written diagonally (only the diagonal has nonzeroes), based on a different basis.
T(v) = A*v = lambda*v is the right relation. the eigenvalues are all the lambdas you find, the eigenvectors are all the v's you find that satisfy T(v)=lambda*v, and the eigenspace FOR ONE eigenvalue is the span of the eigenvectors cooresponding to that eigenvalue.
suppose for an eigenvalue L1, you have T(v)=L1*v, then the eigenvectors FOR L1 would be all the v's for which this is true. the eigenspace of L1 would be the span of the eigenvectors OF L1, in this case it would just be the set of all the v's because of how linear transformations transform one dimension into another dimension. the (entire) eigenspace would be the span of all the eigenvectors from all the eigenvalues.
• Is it always necessary to have row of zeros?
• Yes, otherwise the vector from that matrix wouldn't be an eigenvector.
• I feel I may have missed a video as I don't know what a "reduced row echelon form of a matrix" is and how to make one?
(1 vote)
• Is there a general rule for the relationships of eigenvalues graphically? Eg. Are the eigenvectors of the corresponding eigenvalue perpendicular...?
• The eigenvalues don't have any direction because they're scalars. For some 2x2 matrices the eigenspaces for different eigenvalues are orthogonal, for others not.

An nxn matrix always has n eigenvalues, but some come in complex pairs, and these don't have eigenspaces in R^n, and some eigenvalues are duplicated; so there aren't always n eigenspaces in R^n for an nxn matrix. Some eigenspaces have more than one dimension.
• For E5 shouldnt it be t{2;1] since v1=.05v2 which would mean 2v1=1v2
• Yes, v1 = (v2)/2. What's the solution space (all the vectors v = (v1, v2) | (lambda*I - A)v = 0)? First, what's the free variable in the rref?

(v2 (because it's not a pivot variable - it's not constrained by the solution - it can be any real number). What is v1 in terms of v2?

(v1 is constrained by the solution to be (v2)/2.) What's {v = (v1, v2)} (the entire solution space) in terms of the free variable(s)?

({(v1, v2)} = {((v2)/2, v2) | v2 is a real number}.) How do you express that as a span of basis vectors?

({(v2)/2, v2)} = {v2(1/2, 1)} = span(1/2, 1) = span(1, 2) = {t(1, 2) | t is real}.
• So what if we know what the eigenvectors and eigenvalues are? What practical purpose does this serve? I mean where would you use this?
(1 vote)
• Eigen values and vectors are used extensively in more advanced economics.
• I have a bit of an in-depth question about a practical implication of the use of "eigen-everything". It requires a bit of an introduction, sorry about that.

So, I came here to try to figure out why eigenvalues and eigenvectors are used in the course I'm assisting at university called Theoretical ecology. Most of that course is using differential equations to describe population dynamics between populations of species that influence one another (predator–prey systems etc.). You plot the populations on different axes in a graph, and you get one or more "nullclines" for each equation, which are the lines of all points where the equation equals zero, in other words the population does not change given those population densities for all the populations involved. On either side of such nullclines, the population either grows or declines, and this changes (or in other words, the sign of the value of the differential equation flips) across the nullcline. Such a graph is called the phase space, and you can often times see if some equilibrium (which is where the nullclines intersect -- after all, that's when the growth of all populations equals zero) is stable (perturbations die out) or unstable (perturbations grow) by looking at the sign of the derivatives on all sides of the equilibrium in the phase space.

However, sometimes it's not immediately obvious, and you actually need to calculate the so-called Jacobian matrix, which is the matrix of partial derivatives of the differential equations. E.g., how does the change in population X (which is zero in the equilibrium) change with a small positive step in population density X? How about with a small positive step in population density Y? How about the change in population Y with each of these steps? (That's all of them for a two-dimensional system, giving you a two-by-two Jacobian matrix.)

Now, as it turns out, the equilibrium will then be stable if both eigenvalues of this matrix ('both' if it's two-by-two, anyway—I have yet to look at the next few videos to see if there will be more in larger matrices) are negative. But this is very difficult for me to understand why this is the case intuitively. Can anybody help me with this? What would the eigenvalues or eigenvectors in this case represent in biological terms?

Hopefully this is all a bit understandable for someone who's never considered differential equations for population dynamics before—or, alternatively, there's someone out there who has who can answer this question. Thanks in advance.
• At Sal writes t[ -1, 1] but in the video v2 = -1 and v1 = 1, shouldn't it be t[ 1, -1] ?
• What equation does the N(rref) @ around represent?

(v1 + v2 = 0). What does this equation mean about v1 and v2?

(They can be any real numbers, positive or negative, as long as their sum is 0.)

Which is the pivot variable, v1 or v2?

(Since by convention v1 comes first, it's coefficient is the first non 0 entry in the rref, so it's the pivot variable, and v2 is the free variable.)

What does it mean, that v2 is a free variable?

(v2 can be any real number.)

What's the solution set {(v1, v2) | v1+v2=0} How should it be expressed?

(Since the value of v1 depends on the value of v2, let's express (v1, v2) in terms of v2: {(v1, v2) = (-v2, v2)}.) What is this set as a span of fixed vectors?

{(v1, v2) = (-v2, v2) = v2(-1, 1) = span(-1, 1) = t(-1, 1) for any real t}