What does an eigenvector mean when a matrix transforms it?
Quick answer
An eigenvector is a nonzero vector that a square matrix transforms into a scalar multiple of itself: it stays on the same line through the origin, though its length or orientation may change. The scalar multiple is the eigenvalue. For example, the matrix [[6,5],[1,2]] sends (5,1) to (35,7), which is 7 times (5,1).
The question, the details, the next step.
What does it mean geometrically when a matrix transforms an eigenvector? I understand the equation Mv = λv, but not what I should picture.
Picture the vector as an arrow starting at the origin. A matrix usually moves that arrow to a different direction and length. If the starting vector is an eigenvector, its transformed arrow lies on the same line through the origin as before: the matrix has scaled it, possibly reversing its orientation. The vector is the eigenvector; the number that scales it is the eigenvalue. An eigenvector must be nonzero. [S1]
For instance, if the eigenvalue is 3, the new arrow points the same way and is three times as long. If it is -2, the arrow points the opposite way and is twice as long. So “same direction” is not always quite right: the invariant idea is the same line, with orientation allowed to reverse.
Can you work through a small matrix example and show how to check it?
Take the matrix and vector below:
M = [[6,5],[1,2]], v = (5,1)
Multiply each row of the matrix by the vector:
- First component: 6×5 + 5×1 = 35.
- Second component: 1×5 + 2×1 = 7.
Therefore Mv = (35,7). Now compare that with the original vector: (35,7) = 7(5,1). The output is exactly seven times the original vector, so (5,1) is an eigenvector of this matrix, and its eigenvalue is 7. This example is also given in the cited MIT course notes. [S1]
Geometrically, the arrows (5,1) and (35,7) lie on the same ray from the origin; the second is seven times as long. This calculation establishes the result for this vector and matrix, not for every vector or matrix.
How do I distinguish the eigenvector from the eigenvalue when I’m doing a problem?
Keep track of what kind of thing each symbol represents. The eigenvector v is a nonzero vector, such as (5,1). The eigenvalue λ is a scalar, such as 7. In Mv = λv, the left side is the result of applying the matrix; the right side says that result equals the original vector multiplied by one number.
A useful check is to calculate Mv, then see whether the result equals one scalar times the original vector. In the example, 35/5 and 7/1 are both 7. If the ratios differ, that proposed vector is not an eigenvector for that matrix. Avoid dividing by a component that is zero; instead check the scalar-multiple equation component by component. The zero vector is not an eigenvector, because an eigenvector must be nonzero. [S1]
What’s a reusable way to write up an eigenvector check, and what should I say if the eigenvalue is negative or zero?
Use this short template:
M = [matrix] and v = [nonzero vector].
Mv = [calculated output].
[calculated output] = [scalar]v.
Therefore, v is an eigenvector of M with eigenvalue [scalar].
Geometrically, the output stays on the same line as v; [describe scaling and orientation].
For the worked example, the final line is: “Therefore, (5,1) is an eigenvector with eigenvalue 7. Its image stays on the same line and points the same way, scaled by 7.” A negative eigenvalue means the image stays on the same line but points the opposite way. A zero eigenvalue is possible: it sends the nonzero eigenvector to the zero vector, so there is no resulting arrow orientation to describe. The vector itself must still be nonzero. [S1]
A practical checklist
Confirm the matrix is square and the proposed vector is nonzero.
Multiply the matrix by the vector carefully, one row at a time.
Check whether the output equals one scalar times every component of the original vector.
Name the vector as the eigenvector and the scalar as the eigenvalue.
Describe the geometry as staying on the same line; mention reversed orientation for a negative eigenvalue.
A few more useful answers
Does an eigenvector always keep exactly the same direction?
It stays on the same line through the origin, but a negative eigenvalue reverses its orientation. A positive eigenvalue preserves orientation; a zero eigenvalue sends the vector to the origin.
Can the zero vector be an eigenvector?
No. An eigenvector must be nonzero; otherwise it would not identify a direction.
Does this example show that every vector stays on its original line?
No. It verifies that particular vector for that particular matrix. Other vectors may be sent in different directions; this example does not establish anything about all vectors or matrices.
Sources
References used for this guide.