You are Here: Home >< Maths

# How are invariant lines/planes linked to eigenvalues/vectors? watch

1. Could someone explain the link between invariant lines / planes and eigenvalues/vectors?

Is the eigenvalue the gradient of the invariant line y=mx under a given linear transformation? What about cases where the invariant line is y=mx + c?

What does the eigenvector represent? The unit vector in the direction of the invariant line?

Can finding the eigenvalues/vectors of a 3x3 matrix be used to find the equations of the planes which map to themselves under that matrix?
2. (Original post by SamKeene)
Could someone explain the link between invariant lines / planes and eigenvalues/vectors?

Is the eigenvalue the gradient of the invariant line y=mx under a given linear transformation? What about cases where the invariant line is y=mx + c?

What does the eigenvector represent? The unit vector in the direction of the invariant line?

Can finding the eigenvalues/vectors of a 3x3 matrix be used to find the equations of the planes which map to themselves under that matrix?
eigenvectors

are vectors therefore have "gradient" embodied in them
e.g

i + 2j represents y = 2x
(if y = mx, then m is not the eigenvalue)

they represent invariant directions in 2D and 3D space (or indeed in theoretical 4D+ space)

in 3D space occasionally they represent invariant planes
these is when the 3 equations do not reduce to the usual 2 but they are the same equation. ( I doubt if this is part of A level)
3. (Original post by TeeEm)
eigenvectors

are vectors therefore have "gradient" embodied in them
e.g

i + 2j represents y = 2x
(if y = mx, then m is not the eigenvalue)

they represent invariant directions in 2D and 3D space (or indeed in theoretical 4D+ space)

in 3D space occasionally they represent invariant planes
these is when the 3 equations do not reduce to the usual 2 but they are the same equation. ( I doubt if this is part of A level)
So an eigenvector represents the lines which map onto themselves which pass through the origin.

What about lines which map to themselves which do not pass through the origin? Or is that stepping outside of linear transformations?
4. (Original post by SamKeene)
So an eigenvector represents the lines which map onto themselves which pass through the origin.

What about lines which map to themselves which do not pass through the origin? Or is that stepping outside of linear transformations?
these are not represented by eigenvectors

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: January 29, 2015
Today on TSR

### The most controversial member on TSR?

Who do you think it is...

### Uni strikes! How do they affect you?

Discussions on TSR

• Latest
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE