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

22

matrix

A

applied to a

21

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.