Geometric action of a 3D matrix
in Assignments
(10 points)
In this problem, your mission is to describe the geometric action of a 3D matrix with complex eigenvalues. You can get your matrix in problem #2 from the problems for Monday, March 30 on our online presentations page. Your answer should have several parts:
1) The real eigenvalue,
2) The corresponding real eigenvector,
3) A description of the geometric implications of the real eigenvalue/eigenvector pair for the action of the matrix,
4) The complex eigenvalues listed in the form $a \pm bi$,
5) The corresponding complex eigenvectors, and
6) A description of the geometric implications of the complex eigenvalue/eigenvector pairs for the action of the matrix.
Comments
My Matrix is:
$$A=
\begin{bmatrix}
-4 & 2 & -4 \\
7 & 0 & -5 \\
-3 & -4 & -5
\end{bmatrix}$$
1.) and 2.) I used python to do this:
A = np.array([ [-4,2,-4], [7,0,-5], [-3,-4,-5] ]) eig(A) #out: (array([ 5.47778582+0.j , -7.23889291+0.95112526j, -7.23889291-0.95112526j]), array([[ 0.35370618+0.j , 0.72657829+0.j , 0.72657829-0.j ], [ 0.83561814+0.j , -0.39982124-0.26253258j, -0.39982124+0.26253258j], [-0.42027879+0.j , 0.38841669-0.30403303j, 0.38841669+0.30403303j]]))Giving me a real eigenvalue of $ \lambda = 5.4777882$ and its corresponding eigenvector of: <0.35370618, 0.83561814, -0.42027879 >
Which looks ugly. So I solved the characteristic polynomial by hand:
$$ -\lambda^3-9\lambda^2+26\lambda +292$$ plugging that into wolfram: I get my real answer to be $ \lambda = 5.4778$. So It was just born ugly.
3.) so this pair acts as a description of the matrix's 1D real subspace. what this 1D subspace describes is the stretching or contraction of vector multiplied by this matrix. Since my eigenvalue > 1, this particular matrix has a stretching factor. If it was <1, it would have a contracting nature.
4.) The complex eigenvalues are given by
They are:
$$ -7.23889291 \pm 0.95112526i $$
5.) Their corresponding eigenvectors are:
<0.72657829+0.j, -0.39982124-0.26253258j, 0.38841669-0.30403303j>
and
<0.72657829-0.j,-0.39982124+0.26253258j, 0.38841669+0.30403303j>.
6.) The implication of the complex eigenvalues/vectors:
Their existence implies that this matrix transform vectors by rotation. Where as cos(x) = the real part of the eigenvalue, with x being the angle of rotation that matrix acts upon a vector. Since that was out of arccos range: I can get a non polar answer by
$arccos(-7.23889291) = -40.3375^\degree$: In Polar coordinates: Thing is wildly out of range for arccosine.
Since that was out of arccos range: I can get a nonpolar answer by taking the arctan of the imaginary by real parts of the eigenvalue.
$arctan(0.95112526/-7.23889291) = \theta = -.1306^\degree$
@dan When you use the
eigcommand, the second term is a matrix whose columns are the eigenvectors. You've used the rows.My matrix is
$
A=
\begin{bmatrix}
3 & 0 & 5 \\
3 & 4 & 4 \\
-2 & -4 & 3
\end{bmatrix}
$. As usual I used
numpy.linalg.eig()to compute the eigenvectors and eigenvalues.The real eigenvalue is
1.35306081and its eigenvector isSince the eigenvalue is greater than 1, the subspace of the real vector
$
\begin{bmatrix}
0.813 \\
-0.517 \\
-0.268
\end{bmatrix}
$ will be stretched.The vector also forms the axis of rotation pointing in octant 3 with its inverse pointing in octant 6.
The complex eigenvalues are
4.3234696 +5.34862376j, 4.3234696 -5.34862376jor $4.3234696 \pm 5.34862376i$ when they are formatted.
The corresponding eigenvectors are
or
$
\begin{bmatrix}
-0.130 \pm 0.524i\\
0.262 \pm 0.534i\\
-0.595
\end{bmatrix}
$ when formatted.
The plane being rotated is spanned by the vectors
$
\begin{bmatrix}
-0.130 \\
0.262 \\
-0.595
\end{bmatrix}
$ and
$
\begin{bmatrix}
0.524 \\
0.534 \\
0
\end{bmatrix}
$
which were taken from the complex eigenvectors. The elements in that subspace will be
expanded by the magnitude of the complex eigenvalue $\sqrt{{4.32}^2+{5.35i}^2}$ and rotated in the plane by the angle between the real and imaginary components $tan^{-1}(\frac{5.35}{4.32})$ of the eigenvalue. I calculated this in python with
and the result was
My matrix is,
$$A=\begin{pmatrix}
3&2&-3\\
7&0&3\\
-2&8&-2\end{pmatrix}$$
I used Python to extract the eigenpairs, which returned the eigenvalues,
$$\lambda_{1}\approx-8.29, \lambda_{2}\approx4.65+2.33i, \lambda_{3}\approx4.65-2.33i$$
and the eigenvectors,
$$\vec{v_{1}}\approx\begin{pmatrix}
0.304\\
-0.541\\
0.784\end{pmatrix}, \vec{v_{2}}\approx\begin{pmatrix}
-0.0173+0.380i\\
0.558+0.292i\\
0.677\end{pmatrix}, \vec{v_{3}}\approx\begin{pmatrix}
-0.0173-0.380i\\
0.558-0.292i\\
0.677\end{pmatrix}.$$
The real eigenpair indicates that there is a vector equal to the eigenvector $\langle0.304,-0.541,0.784\rangle$ about which the rotation will occur. This vector will be stretched in the opposite direction based on the magnitude and sign of the eigenvalue.
The complex eigenpair indicates that the plane spanned by the real part of the eigenvector, $\langle-0.0173,0.558,0.677\rangle$ and the imaginary part $\langle0.38,0.292,0\rangle$ will rotate about the vector described above. In order to find the angle of rotation, I normalized the complex eigenvalue so it would be in range of arccos and arcsin,
$$\frac{2.33}{\sqrt{2.33^{2}+4.65^{2}}}\approx0.448 \text{ and } \frac{4.65}{\sqrt{2.33^{2}+4.65^{2}}}\approx0.894,$$
which results in $\cos(1.11)\pm i\sin(1.11)$. Lastly, the magnitude of the complex eigenvalues indicate the stretching on the plane will occur.
Python code:
import numpy as np from scipy.linalg import eig A = np.matrix([ [3,2,-3], [7,0,3], [-2,8,-2] ]) vals, vecs = eig(A) print('eigenvalues =', vals) print('') print('eigenvectors =', vecs) Out: eigenvalues = [-8.29030743+0.j 4.64515371+2.33282053j 4.64515371-2.33282053j] eigenvectors = [[ 0.30418007+0.j -0.01729465+0.38002567j -0.01729465-0.38002567j] [-0.54065897+0.j 0.55804045+0.29242767j 0.55804045-0.29242767j] [ 0.78432286+0.j 0.67702164+0.j 0.67702164-0.j ]]My matrix was
$A=\left[\begin{matrix}
3 & 5 & 1 \\
-2 & 8 & 0 \\
-4 & 6 & 4
\end{matrix}\right]$
To begin, I had to load in the tools!
Next was to determine the eigenvectors and values, I did this by
A=np.matrix([ [3,5,1], [-2,8,0], [-4,6,4] ]) evals, evecs = eig(A) print("Evals = ", evals) print('---') print("Evecs = ", evecs)Which produced the following eigenvalues:
$\lambda_1 = 5.27396414 + 2.68796746j \\
\lambda_2 = 5.27396414-2.68796746j \\
\lambda_3 = 4.45207171+0.j
$
For the real eigenvalue $\lambda_3 = 4.45207171$ we can expect a stretch of 4.4x the original vector.
The corresponding vector that describe this stretch is
$ \vec{\lambda_3} = \left[\begin{matrix}
-0.5603 \\
-0.316\\
0.7657 \\
\end{matrix}\right] $
The other two values are complex values that indicate an expansion and rotation, these vectors are
$\vec{\lambda_{1 and 2}} = \left[\begin{matrix}
-0.67828 \\
-0.25231\\
-0.28081\\
\end{matrix}\right]
\pm
\left[\begin{matrix}
0 \\
0.24879\\
0.57924\\
\end{matrix}\right] i $
The magnitude of this stretch in the complex plane is given by
$\sqrt{5.27^2 + 2.69^2} = 5.92$
and its rotation by
$tan^{-1}(\frac{2.69}{5.27}) = 0.472$
My matrix is,
$$
A = \begin{pmatrix}
-1 & -1 & 2\\
-1 & -2 & -2\\
-3 & -2 & -4
\end{pmatrix}
$$
I determined the eigenvalues and eigenvectors using Python,
import numpy as np from scipy.linalg import eig, inv A = np.array([ [-1, -1, 2], [-1, -2, -2], [-3, -2, -4] ]) evals,evecs = eig(A) print(evals) print() print(evecs) # out [-4.24189656+0.j -1.37905172+1.18263556j -1.37905172-1.18263556j] [[-0.30675473+0.j 0.72950896+0.j 0.72950896-0.j ] [ 0.55375979+0.j -0.43580135-0.21428757j -0.43580135+0.21428757j] [ 0.77411345+0.j -0.35616149+0.32422783j -0.35616149-0.32422783j]]So we have,
$${\lambda}_R = -4.242 \ \ , \ \ \ {\lambda}_C = -1.379 \pm 1.183i$$
$$\vec{v}_R = \begin{bmatrix}
-0.307\\
0.554\\
0.774
\end{bmatrix} \ \ , \ \ \ \vec{v}_C = \begin{bmatrix}
0.730\\
-0.436 \mp 0.214i\\
-0.356 \pm 0.324i
\end{bmatrix}$$
The real eigen-pair is,
$${\lambda}_R = -4.242 \ , \ \ \vec{v}_R = \begin{bmatrix}
-0.307\\
0.554\\
0.774
\end{bmatrix} \\$$
Geometrically, this means that any vector in the 1D subspace $span(\vec{v}_R)$ remains in $span(\vec{v}_1)$ (no rotation or skewing) and is scaled by ${\lambda}_R = -4.242$ (reflected about the origin and scaled by 4.242) under the transformation.
The complex eigen-pairs are,
$$ {\lambda}_{C1} = -1.379 + 1.183i $$
$$\vec{v}_{C1} = \begin{bmatrix}
0.730\\
-0.436-0.214i\\
-0.356+0.324i
\end{bmatrix} = \begin{bmatrix}
0.730\\
-0.436\\
-0.356
\end{bmatrix} + i \begin{bmatrix}
0\\
-0.214\\
0.324
\end{bmatrix} = \vec{v}_a + i \vec{v}_b \\$$
$\\$
$${\lambda}_{C2} = -1.379 - 1.183i$$
$$\vec{v}_{C2} = \begin{bmatrix}
0.730\\
-0.436+0.214i\\
-0.356-0.324i
\end{bmatrix} = \begin{bmatrix}
0.730\\
-0.436\\
-0.356
\end{bmatrix} + i \begin{bmatrix}
0\\
0.214\\
-0.324
\end{bmatrix} = \vec{v}_a - i \vec{v}_b$$
$\\$
Since complex eigenvalues exist we know that there is a rotation, $\lambda = cos(\theta) \pm i sin(\theta)$. This rotation takes place in the plane spanned by $\vec{v}_a$ and $\vec{v}_b$. The angle of rotation can be found from the equation relating $\lambda$ to $\theta$,
$$tan(\theta) = \frac{sin(\theta)}{cos(\theta)} = \frac{-1.379}{-1.183} \ \longrightarrow \ \theta = tan^{-1} \bigg{(} \frac{1.379}{1.183} \bigg{)} = \boxed{40.6^{\circ}}$$
Along with the rotation, scaling also occurs in this 2D subspace, the scaling factor is $\sqrt{(1.379)^2 + (1.183)^2} = \boxed{1.820}$ .
I was assigned the following matrix:
$$
A = \left(
\begin{array}{cccc}
-5 & 5 & 8 \\
0 & -5 & -4 \\
7 & 8 & -3 \\
\end{array}
\right).
$$
I then used the following code to determine the eigenvalues and eigenvectors:
import numpy as np #import math as mt from scipy.linalg import lu, inv, solve, lu_solve, lu_factor, norm, eig, inv A = np.array([ [-5, 5, 8], [0, -5, -4], [7, 8, -3] ]) eigvals, eigvecs = eig(A) print('\n Eigenvalues:') print(eigvals) print('\n Eigenvectors:') print(eigvecs)The following is the output of the code:
From here, it is seen that the real eigenvalue, $ \lambda_{real} = -10.963 $. This result means that the subspace of the corresponding vector will be stretched by a factor of 10.963 in the opposite direction of the vector.
The real eigenvector is the following:
$$ v_{real} = \left(
\begin{array}{cccc}
0.84518497 \\
-0.29775284 \\
-0.44385315 \\
\end{array}
\right).
$$
The complex eigenvalues are $\lambda_{complex} = -1.019 \pm 2.762j $. The corresponding eigenvectors are described by the following matrix:
$$ v_{complex} = \left(
\begin{array}{cccc}
0.63570305 \\
0.43769115 \pm 0.22343445j \\
-0.58992674 \pm 0.07982215j \\
\end{array}
\right).
$$
In order to describe the geometric implications, the magnitude and direction of the complex eigenvalue must be found:
$$ x_{magnitude}^2 = (-1.019)^2 +(2.762)^2 $$
$$ x_{magnitude} = 2.944 $$
This means the vector will stretch by a factor of 2.944. The amount of rotation can be found by taking the inverse tan of the real and imaginary components to find the angle between the axes. Since there is a +/- in front of the imaginary component, there will be two angles with equal magnitude but flipped along the real axis:
$$ cos(\theta) = \frac{-1.019}{2.944} = -0.346 $$
$$ sin(\theta) = \frac{\pm 2.762j}{2.944} = \pm 0.938j $$
$$ \theta = tan^{-1}(\frac{ \pm 2.762}{-1.019}) = \pm 1.217 radians$$
This results indicates that the angle of the rotation that will occur will be $ \pm 1.217 $ radians from the origin.
To begin this problem, I loaded the necessary libraries into my Jupyter Notebook and defined my matrix $A$.
import numpy as np from scipy.linalg import eig A = np.matrix([ [8,8,1], [5,1,-2], [-5,5,4] ])I then used the
eig()command to compute the eigenvalues and eigenvectors for the matrix.eVals, eVecs = eig(A) print("Eigenvalues = \n", eVals) print("Eigenvectors = \n", eVecs)This gave
$\lambda_1 = 11.838$
$\lambda_2 = 0.581 + 2.214i$
$\lambda_3 = 0.581 - 2.214i$
Corresponding to $\lambda_1$, we have the eigenvector $v_1 = \begin{pmatrix} 0.857 \\ 0.444 \\ -0.263 \end{pmatrix}$
Geometrically, then, multiplication of $v_1$ by $A$ will stretch the vector by $\lambda_1$, such that $ A \cdot v_1 = \lambda_1 \cdot v_1$.
The other eigenpairs are complex, so the geometric action will be comprised of an extension and a rotation.
$v_2 = \begin{pmatrix} 0.274 \\ -0.318 \\0.865\end{pmatrix} + \begin{pmatrix} -0.159 \\ 0.223 \\ 0 \end{pmatrix}i$
$v_3 = \begin{pmatrix} 0.274 \\ -0.318 \\0.865\end{pmatrix} - \begin{pmatrix} -0.159 \\ 0.223 \\ 0 \end{pmatrix}i$
This transformation takes place in the span of ${v_2,v_3}$.
The magnitude of this stretch is given by
and the angle of rotation, by
For this problem, I got the matrix
$$ A = \begin{pmatrix}
4 & -3 & -4\\
8 & 0 & -5\\
3 & 3 & -1
\end{pmatrix}.$$
I started with using the following Python code to find all of the eigenvalues and eigenvectors:
A = np.matrix([ [4,-3,-4], [8,0,-5], [3,3,-1] ]) eig(A) Out: (array([ 1.65613828+6.72989987j, 1.65613828-6.72989987j, -0.31227655+0.j ]), array([[-0.34237548-0.38508407j, -0.34237548+0.38508407j, 0.50991797+0.j ], [-0.71808403+0.j , -0.71808403-0.j , -0.32757731+0.j ], [-0.30995147+0.35039222j, -0.30995147-0.35039222j, 0.79540981+0.j ]]))1.
Python tells us that our real eigenvalue $\lambda_3=-.3123$.
2.
From this, we can also see that our real eigenvector, $v_3 = <.5099, -.3276, .7954>$.
3.
This implies that our vector $v_3$ will contract in the opposite direction, since our real eigenvalue is both negative and it's absolute value is less than 1.
4.
We can see that our complex eigenvalues are as follows:
$\lambda_1=1.6561 + 6.7299i$,
$\lambda_2 = 1.6561 - 6.7299i$.
This is as expected, as complex eigenvalues come in conjugate pairs.
5.
The corresponding eigenvectors are then
$v_1 = <-.3424-.3851i, -.7181+0i, -.3100+.3504i>$,
$v_2 = <-.3424+.3851i, -.7181+0i, -.3100-.3504i>$.
6.
This tells us that our geometric transformation will take place in the span of $v_1$ and $v_2$, with both an expansion and rotation.
The expansion can be found by finding the magnitude of our complex eigenvalue. We find that
$$v_{magnitude} = \sqrt{(1.6561)^2+(6.7299)^2} = 6.9307.$$
So this means our vector stretches by a factor of 6.9307.
To find the rotation, we just use
$$\theta = tan^{-1}(\frac{6.7299}{1.6561})= 1.3295 radians.$$
So our vector has a rotation of 1.3295 radians, or approximately $76.1753^o$.
So first off my matrix was
I started by calculating the characteristic polynomial by hand.
Next I used the following python code to calculate the eigen values and eigen vectors.
This yielded the real eigen value, -8.0269, and the two imaginary eigen values, 0.5134+1.4067i and 0.5134-1.4067i.
The corresponding eigen vectors are
respectively.
These eigen values represent the magnitude of the stretch in the direction of their corresponding eigen vectors.
For this problem, my matrix was given as
$$
M =
\begin{bmatrix}
2 & 7 & 3 \\
-4 & 3 & 4 \\
7 & 2 & -5
\end{bmatrix}.
$$
To ascertain the geometric action this matrix performs, I used python to look at the eigenvalues and eigenvectors of the matrix.
M = array([[2,7,3],[-4,3,4],[7,2,-5]]) evals,evecs = eig(M) print(' evals=',evals,'\n','evecs=\n',evecs) #output evals= [-5.77375223+0.j 2.88687611+2.23655116j 2.88687611-2.23655116j] evecs= [[-0.01592289+0.j 0.73764017+0.j 0.73764017-0.j ] [ 0.40875635+0.j -0.15822615+0.27695403j -0.15822615-0.27695403j] [-0.91250463+0.j 0.58725949-0.09630275j 0.58725949+0.09630275j]]There is one real eigenvalue and a conjugate pair of complex eigenvalues, so there must be some combination of stretching and rotation.
Addressing the real eigenvalue first, there must be an expansion of $~5.77$ in the direction of $\vec{v}=$
-evecs[0]or$$
\vec{v}=
\begin{bmatrix}
0.01592289 \\
-0.40875635\\
0.91250463
\end{bmatrix}.
$$
Moving on there are 3 important bits of information that can be extracted from the complex eigenvalue pair. They provide an expansion, an angle of rotation and their eigenvector defines the plane in which these occur. Python is again used to determine these.
mag = sqrt(evals[1].real**2+evals[1].imag**2) T = arctan(evals[1].imag/evals[1].real) print('expansion in the plane :',mag) print('angle of rotation :',T,'rad') S_u = evecs[:,1].real S_v = evecs[:,1].imag print(' S_u=',S_u,'\n','S_v=',S_v) #output expansion in the plane : 3.651878257926536 angle of rotation : 0.6591417152887815 rad S_u= [ 0.73764017 -0.15822615 0.58725949] S_v= [ 0. 0.27695403 -0.09630275]Therefore there must be an expansion of $~3.65$ and a rotation of $~0.659rad$ in the plane spanned by $(S_u,S_v)$.
My matrix is
$$A= \begin{bmatrix}
0 & 3 & 3 \
1 & -4 & 0 \
4 & 8 & -4
\end{bmatrix}$$
Using
import numpy as np from scipy.linalg import eig A = np.matrix([ [0,3,3], [1,-4,0], [4,8,-4] ]) vals, vecs = eig(A) print('eigenvalues =', vals) print('') print('eigenvectors =', vecs)I got eigenvalues $\lambda_1 = 2.75, \lambda_2 = -5.37 + 1.29i, \lambda_3 = -5.37 - 1.29i$
with eigenvectors $v_1 = [0.79, 0.12, 0.61], v_2 = [0.44+0.25i, -0.08-0.26i, -0.82+0i], v_3 = [0.44-0.25i, -0.08+0.26i, -0.82-0i]$ respectively.
This implies an expansion of 5.53 and a rotation of 0.24 radians.
@eli You wrote:
I do think there's a bit more to it than this. Your complex eigenvalues have absolute value 5.53 and argument $\pm(\pi-0.24)$ so what you've written is certainly relevant. That effect happens in the plane spanned by the vectors
$$\langle 0.44,-0.08,-0.82 \rangle \: \text{ and } \: \langle 0.25,-0.26, 0\rangle.$$
In addition, the real eigenvalue implies a stretch through the factor 2.75 along the line spanned by the real eigenvector.