WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

Vector Transformations and Eigenvectors of 2×2 Matrix

​
matrix A =
a
b
c
d
=
1
2
4
7
a
b
c
d
vector x =
j
k
=
5
6
j
k
color
determinant A =
-1
inverse A =
-7
2
4
-1
first eigenvalue × first eigenvector
λ
1
×
y
1
=
1
4
-3+
17
4+
17
4+
17
second eigenvalue × second eigenvector
λ
2
×
y
2
=
1
4
-3-
17
4-
17
4-
17
matrix vector product A·x = b
(​1×5​)+(​2×6​)
(​4×5​)+(​7×6​)
=
17
62
determinant A calculation
(​1×7​)-(​2×4​)
inverse A calculation
7
-2
-4
1
(1/(​(​1×7​)-(​2×4​)​)​)
eigenvalue λ
eigenvector y
4+
17
1
4
(-3+
17
)
1
4-
17
1
4
(-3-
17
)
1
matrix A moves black vector x to b
This Demonstration shows the transformation represented by a
22
matrix
A
applied to a
21
vector
x
. The vector
x
is transformed to a new vector
b
, shown in color. If the vector
x
is an eigenvector of
A
, then
b
is simply scaled by
λ
, the eigenvalue, without changing direction (except the direction is reversed if
λ<0
.) Color-coded formulas show the calculation of the inverse, determinant and new vector
b
, as well as the eigenvectors of
A
.
If matrix
A
is singular and has no inverse, it is indicated.
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.