fx-991 CW: Finding the Eigenvalues of a 2 x 2 Matrix
Finding
the Eigenvalues of a 2 x 2 Matrix
To
find the eigenvalues of a matrix M, the characteristic polynomial
must be solved in terms of λ:
det(
M – λ * I) = 0
where:
M
= a square matrix of dimension n x n
I
= an identity matrix of size n x n.
The
identity matrix is a square matrix in which all elements have a value
0 except the diagonal elements, which have the value 1.
A
2 x 2 identity matrix: [ [ 1, 0 ] [ 0, 1 ] ]
A
3 x 3 identity matrix: [ [ 1, 0, 0 ] [ 0, 1, 0 ] [ 0, 0, 1 ] ]
For
a 2 x 2 matrix:
M
= [ [ a, b ] [ c , d ] ]
The
characteristic polynomial used to find the eigenvalues are:
det(
M – λ * I) = 0
det(
[ [ a, b ] [ c , d ] ] - λ * [ [ 1, 0 ] [ 0, 1 ] ] ) = 0
det(
[ [ a, b ] [ c , d ] ] - [ [ λ, 0 ] [ 0, λ ] ] ) = 0
det(
[ [ a - λ, b ] [ c , d - λ ] ] ) = 0
(a
– λ) * (d – λ) – b * c = 0
λ^2
– (a + d) * λ + (a * d – b * c) = 0
Note:
trace(M)
= a + d
det(M)
= a * d – b * c
The
characteristic equation to be solved is:
λ^2
– (a + d) * λ + (a * d – b * c) = 0
λ^2
– trace(M) * λ + det(M) = 0
Casio
fx-991CW Algorithm
The
algorithm will involve two apps: Matrix and Equation. Here
is a way of finding the eigenvalues of a 2 x 2 matrices using only
the fx-991 CW calculator without the need for writing anything down.
Note:
The variables I use in the procedure is just for illustrative
purposes. You can use any variables you want to designate the corner
elements, the trace, and the determinant. The point is to be
organized.
Variables
used in this procedure:
A
= upper-left element
B
= lower-right element
C
= trace = A + B
D
= determinant
Settings:
It is assumed that the MathI/MathO Input/Output mode and a+bi is
selected for Complex result.
The screen shots are generated
using the ClassPad Math (classpad.net) emulator for the fx-991CW and
illustrates finding the eigenvalues of the matrix:
MatA = [ [ 4, 2 ] [ 5, 4 ] ]
Matrix
App
Step
1: Press the [
HOME ]
key, select the Matrix app.
Step
2: Use the TOOLS key to
define a Matrix of dimension 2 x
2.
Step
3: Enter the matrix’s
elements. Register each element is registered by using [ OK ] or [
EXE ]. Be careful not to press [ EXE ] or [ OK ] without entering a
value first, as we need to keep the matrix editing screen up.
Step
4: Go to element (1,1) (upper-left hand corner), press the [
VARIABLE] key, go to variable A, press [ OK ], and select Store.
Then go to element (2,2) (lower-right hand corner), press the [
VARIABLE] key, go to variable B, press [ OK ], and select Store.**
Step
5: Without entering or editing an element, press either [ EXE ] or [
OK ] to leave the matrix editor. When the message “Press [TOOLS]
to define Matrix.” appears, we are in the calculation mode of the
Matrix app.
Note:
If you leave the matrix edit mode before storing the corner
elements, you can go back into matrix edit mode by pressing [ TOOLS
], selecting your matrix, and then selecting Edit. You can check to
see if corner values are stored by pressing [ VARIABLE ].
Step
6: Press [ SHIFT ] [ 4 ] (A) + [ SHIFT ] [ 7 ] ( B ) [ EXE ].
This calculates the matrix’s trace. Then press [ VARIABLE ],
choose C, press [ OK ], choose Store.
Step
7: Press [ CATALOG ], select the Matrix sub-menu, then the Matrix
Calc sub-menu, and select Determinant. Then use the catalog to
grab the matrix and press [ EXE ] to calculate the matrix. Use the
variable list to store the value in D (similar procedure in Step 6).
When
transferring between apps, the values stored in the memory registers
A-F, x, y, and z are retained, even when the fx-991CW is turned off.
Equations
App
Step
8: Press [ HOME ] and select the Equation app and press [ OK ].
Select Polynomial, ax²+bx+c (2nd
order polynomial, quadratic equation).
Step
9: Enter the following coefficients: 1 x² – C x + D (note the
minus sign on C).
Step
10: Press [ OK ]. The first eigenvalue is displayed. Press the
down arrow ([↓]) to get the other eigenvalue.
An optional
step is to use the [ VARIABLE ] key to store the results (like in E
or F, for example).
Another optional step is to press [ FORMAT ], select
Decimal to see the decimal approximation.
The
results are:
C
= trace = 8
D
= determinant = 6
Eigenvalues:
λ1
= 4 + √10
λ2
= 4 - √10
Other
Examples
Find
the eigenvalues of Mat B = [
[ -8, 1 ] [ 16, 7 ] ]
Results:
C
= trace = -1
D
= determinant = -72
Eigenvalues:
λ1
= 8
λ2
= -9
Find
the eigenvalues of Mat C = [ [ -5, -7 ] [ 3, - 2] ]
Results:
C
= trace = -7
D
= determinant = 31
Eigenvalues:
λ1
= (-7 + 5 * √3 * i) / 2
λ2
= (-7 - 5 * √3 * i) / 2
I
hope you find this useful and beneficial. Until next time,
Eddie
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original content copyright, © 2011-2025. Edward Shore.
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