Eigenvalues and eigenvectors defined by Av=λv, finding eigenvalues from det(A−λI)=0
The eigenvectors of a matrix A are the vector(s) v such that
for some constant(s) λ which we call eigenvalues.
Notice that any multiple of an eigenvector is also an eigenvector:
The eigenvalues λ of a matrix A satisfy
For example, if A=(−1−234) then
which simplifies to
We call λ2−3λ+2 the characteristic polynomial of A.
If we know an eigenvalue of a matrix A, we can find the corresponding eigenvector using its definition:
When the matrix A is known, we can solve this system of simultaneous equations to find the eigenvector (xy).
The real power of eigenvectors is that they can be used to calculate matrix powers more easily. Matrix multiplication is slow and tedious, but the action of a matrix on its eigenvectors is much simpler.
To see why this it useful, consider this:
For eigenvector v1 and v2 with eigenvalues λ1 and λ2 of a matrix A:
We can write any vector as a combination of these:
Then we can apply An:
If a matrix A has two distinct, real eigenvalues, then we can write it in the form
where P=(v1v2) is formed with the eigenvectors of A as its columns, and D=(λ100λ2) is a diagonal matrix.
Example
A matrix has eigenvectors (32) and (11) with corresponding eigenvalues 1 and 2. Then
and
We can find:
So the matrix is
This is the real power and beauty of eigenvectors.
Essentially, when 2 real eigenvectors exist, the matrix P−1 transforms any vector x to a new coordinate system where the axes are the eigenvectors. Once in this coordinate system, the matrix A simply becomes a stretch of the form (λ100λ2), which is easy to do, and also easy to find powers of.
Once we've done our transformations, we need to get back to our original coordinate system, so we multiply by P. All this means that A is the composition of 3 transformations:
P−1
D=(λ100λ2)
P
Thus
The following animation shows how eigenvectors become the axes. The process of diagonalization is essentially changing the basis vectors to use the eigenvectors instead of (10) and (01).
Nice work completing Eigenvalues & Eigenvectors, here's a quick recap of what we covered:
Exercises checked off
Eigenvalues and eigenvectors defined by Av=λv, finding eigenvalues from det(A−λI)=0
The eigenvectors of a matrix A are the vector(s) v such that
for some constant(s) λ which we call eigenvalues.
Notice that any multiple of an eigenvector is also an eigenvector:
The eigenvalues λ of a matrix A satisfy
For example, if A=(−1−234) then
which simplifies to
We call λ2−3λ+2 the characteristic polynomial of A.
If we know an eigenvalue of a matrix A, we can find the corresponding eigenvector using its definition:
When the matrix A is known, we can solve this system of simultaneous equations to find the eigenvector (xy).
The real power of eigenvectors is that they can be used to calculate matrix powers more easily. Matrix multiplication is slow and tedious, but the action of a matrix on its eigenvectors is much simpler.
To see why this it useful, consider this:
For eigenvector v1 and v2 with eigenvalues λ1 and λ2 of a matrix A:
We can write any vector as a combination of these:
Then we can apply An:
If a matrix A has two distinct, real eigenvalues, then we can write it in the form
where P=(v1v2) is formed with the eigenvectors of A as its columns, and D=(λ100λ2) is a diagonal matrix.
Example
A matrix has eigenvectors (32) and (11) with corresponding eigenvalues 1 and 2. Then
and
We can find:
So the matrix is
This is the real power and beauty of eigenvectors.
Essentially, when 2 real eigenvectors exist, the matrix P−1 transforms any vector x to a new coordinate system where the axes are the eigenvectors. Once in this coordinate system, the matrix A simply becomes a stretch of the form (λ100λ2), which is easy to do, and also easy to find powers of.
Once we've done our transformations, we need to get back to our original coordinate system, so we multiply by P. All this means that A is the composition of 3 transformations:
P−1
D=(λ100λ2)
P
Thus
The following animation shows how eigenvectors become the axes. The process of diagonalization is essentially changing the basis vectors to use the eigenvectors instead of (10) and (01).
Nice work completing Eigenvalues & Eigenvectors, here's a quick recap of what we covered:
Exercises checked off