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
All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
