Eigenvalues calculator

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3131313131351515151515≈83137
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  About matrix eigenvalues calculator

This is a free online matrix eigenvalues calculator with complete, detailed, step-by-step description of solutions, that performs operations with matrices up to 99x99 in size with matrix elements of this type: decimal numbers, fractions, complex numbers, variables.

To start the calculation, you need to first enter the size of the matrix in the input field that you can find from the very top of the screen, also there you can choose the desired method of calculation.

A little below you will find a matrix window in which you need to enter matrix elements using the keyboard. The matrix control panel is also located here, which simplifies work with matrices and contains the following control elements:

  • The first element allows you to expand the matrix window. This can be especially useful in cases where you need to perform calculations with very large matrices that do not fit completely. If the matrix is still not visible after expanding the window, you can change the scale of the matrix using the + / - buttons;
  • The second element performs the function of copying the matrix input to the memory buffer. This can be useful in cases where you often use the same matrix for calculations, or if you need to move matrices between operations;
  • And the last element inserts the previously copied matrix, which allows you to speed up the process of entering the matrix to just a few clicks, instead of doing it manually;

And further down you will find a toolbar that allows you to customize the calculator and make it easier to work with it. It is visually divided into three parts, each of which is responsible for the following functionality:

  • The first allows you to select the number format when the solution result is displayed. Also, here you can turn off comments to the solution of the problem if you have already understood how to solve this problem, and you use the calculator to speed up or check your own calculations. Or you can turn off the step-by-step solution entirely if you only need the result of the solution;
  • The second contains buttons that allow you to change the type of the matrix input field, erase its elements or the entire matrix, and the largest button with an equal sign, which will take you to the screen with the solution of the problem. All these buttons are duplicated by keys on the keyboard. To find out which key on the keyboard to press, simply hover over one of the buttons and a tooltip will appear with the name of the key. You can also use the arrow keys on your keyboard to move the cursor between matrix input fields;
  • And the last one allows you to choose the number of digits after the decimal point for rounding non-integer numbers. Also, here you can immediately see an example of how rounded fractions will look;

  What is eigenvalues of a matrix?

The definition of eigenvalues is closely related to eigenvectors. Eigenvectors are vectors whose directions are not changed by the linear transformation but are scaled by a constant factor, and this constant factor by which the eigenvectors are scaled during the linear transformation is the eigenvalue.

  How to find the eigenvalues of a matrix?

First, we need to find the characteristic equation of the given matrix, and then solve it. The roots of the characteristic equation of a given matrix are also the eigenvalues of this matrix. The eigenvalues of square matrices can only be calculated.

  Example of finding eigenvalues of a matrix

Write the initial matrix
A
:
A
=
71
7
2
4
8
8
5
5
5
5
8
5
2
2
7
2
To find the eigenvalues of the matrix
A
, need to do the following:
1)
Find the characteristic equation of the matrix A, for this need to do the following:
Form a new matrix(A - λI) by subtracting λ from all elements of the main diagonal of the matrix A;
Find the determinant of the matrix A - λI;
Equate the determinant of the matrix A - λI to zero;
2)
Solve the characteristic equation of the matrix A;
3)
The roots of the characteristic equation of the matrix A are also its eigenvalues;
2
Form A − λ·I
Form the matrix
A - λI
:
A - λI
=
A
-
λ
*
I
=
71
7
2
4
8
8
5
5
5
5
8
5
2
2
7
2
-
λ
*
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
=
71
-
λ
7
2
4
8
8
-
λ
5
5
5
5
8
-
λ
5
2
2
7
2
-
λ
Now need to find the determinant of this matrix;
3
Characteristic polynomial det(A − λ·I)
det(
A - λI
) =
71
-
λ
7
2
4
8
8
-
λ
5
5
5
5
8
-
λ
5
2
2
7
2
-
λ
=
λ
4
0
-89
λ
3
0
+
1230
λ
2
0
-1550
λ
-3648
;
4
Characteristic equation
We found the following determinant of the matrix
A - λI
:
λ
4
0
-89
λ
3
0
+
1230
λ
2
0
-1550
λ
-3648
;
Equate this determinant to zero and obtain the characteristic equation of the matrix
A
:
λ
4
0
-89
λ
3
0
+
1230
λ
2
0
-1550
λ
-3648
= 0;
Now we can solve this equation and its roots will give us the eigenvalues of the matrix
A
;
5
Solution of the characteristic equation
Write the initial equation of the roots of which must be found:
λ
4
0
-89
λ
3
0
+
1230
λ
2
0
-1550
λ
-3648
= 0;
As we can see from the equation the maximum degree of the variable is
4
, which means that we have the equation of the following type:
aλ
4
0
+
bλ
3
0
+
cλ
2
0
+
dλ
+
e
= 0;
// where
a
=
1
;
b
=
-89
;
c
=
1230
;
d
=
-1550
;
e
=
-3648
;
To solve this equation, we can use the Ferrari's method, which involves bringing the initial equation to a depressed quartic form;
Depressed quartic form means removal from the equation
λ
3
0
and has the following form:
t
4
0
+
pt
2
0
+
qt
+
r
= 0;
// where
p
=
8
c
-
3
b
2
0
8
;
q
=
b
3
0
-
4
bc
+
8
d
8
;
r
=
-3
b
4
0
+ 256
e
- 64
bd
+
16
b
2
0
c
256
;
Also, if
a
is not equal to
1
, then before bringing the equation to a depressed quartic form, it is necessary to divide all the coefficients of the equation by
a
and before that save the values of
a
and
b
in the variables
aOrigin
and
bOrigin
, as later we will need these values to solve the equation:
aOrigin
=
a
;
bOrigin
=
b
;
a
=
a
a
;
b
=
b
a
;
c
=
c
a
;
d
=
d
a
;
e
=
e
a
;
Next, according to the Ferrari's method, it is necessary to find the following cubic equation equivalent the equation of a depressed quartic form:
m
0
1
y
3
0
+
m
0
2
y
2
0
+
m
0
3
y
+
m
0
4
= 0;
// where
m
0
1
= 1;
m
0
2
=
p
2
;
m
0
3
=
p
2
0
- 4
r
16
;
m
0
4
= -
q
2
0
64
;
Now need to solve the resulting cubic equation, for example, by the Cardano's method;
// where
y
0
1
,
y
0
2
,
y
0
3
is the roots of the cubic equation;
Finally we can find the roots
λ
0
1
,
λ
0
2
,
λ
0
3
,
λ
0
4
of the initial equation:
λ
0
1
=
P
+
Q
+
R
-
S
;
λ
0
2
=
P
-
Q
-
R
-
S
;
λ
0
3
= -
P
+
Q
-
R
-
S
;
λ
0
4
= -
P
-
Q
+
R
-
S
;
// where
P
=
y
0
1
;
Q
=
y
0
3
;
R
= -
q
8
PQ
;
S
=
bOrigin
4
aOrigin
;
This is a general formula for
y
0
1
> 0
and
y
0
3
> 0
, special cases of the formula are described below;
6
Special cases of the formula
y
0
1
> 0
and
y
0
2
= 0
and
y
0
3
= 0:
P
=
y
0
1
;
Q
= 0;
R
= 0;
y
0
1
= 0
and
y
0
2
> 0
and
y
0
3
> 0:
P
=
y
0
2
;
Q
=
y
0
3
;
R
= -
q
8
PQ
;
y
0
1
> 0
and
y
0
2
> 0
and
y
0
3
= 0:
P
=
y
0
1
;
Q
=
y
0
2
;
R
= -
q
8
PQ
;
In the following cases the equation will have non-real complex conjugate roots;
If
y
0
2
and
,
y
0
3
are complex numbers, or
y
0
2
< 0
and
y
0
3
< 0
:
P
=
y
0
2
;
Q
=
y
0
3
;
R
= -
q
8
PQ
;
y
0
1
> 0
and
y
0
2
< 0
and
y
0
3
= 0:
P
=
y
0
1
;
Q
=
y
0
2
;
R
= -
q
8
PQ
;
For each case:
S
=
bOrigin
4
aOrigin
;
7
Depressed quartic form
Divide each coefficient by a:
aOrigin
=
a
=
1
;
bOrigin
=
b
=
-89
;
a
=
a
a
=
1
1
=
1
;
b
=
b
a
=
-89
1
=
-89
;
c
=
c
a
=
1230
1
=
1230
;
d
=
d
a
=
-1550
1
=
-1550
;
e
=
e
a
=
-3648
1
=
-3648
;
Now we can find the coefficients of the equation of the depressed quartic form:
p
=
8
c
-
3
b
2
0
8
=
8 *
1230
- 3 *
-89
2
0
8
=
-1740
3
8
;
q
=
b
3
0
- 4
bc
+
8
d
8
=
-89
3
0
- 4 *
-89
*
1230
+ 8 *
-1550
8
=
-34936
1
8
;
r
=
-3
b
4
0
+ 256
e
- 64
bd
+ 16
b
2
0
c
256
=
-3 *
-89
4
0
+ 256 *
-3648
- 64 *
-89
*
-1550
+ 16 *
-89
2
0
*
1230
256
=
-164469
67
256
;
Depressed quartic form
:
t
4
0
-1740
3
8
t
2
0
-34936
1
8
t
-164469
67
256
= 0;
8
Cubic equation
m
0
1
= 1;
m
0
2
=
p
2
=
-1740
3
8
2
=
-870
3
16
;
m
0
3
=
p
2
0
- 4
r
16
=
-1740
3
8
2
0
- 4 *
-164469
67
256
16
=
230423
57
64
;
m
0
4
= -
q
2
0
64
= -
-34936
1
8
2
0
64
=
-19070825
52
109
;
Cubic equation
:
y
3
0
-870
3
16
y
2
0
+
230423
57
64
y
-19070825
52
109
= 0;
Solve this equation by the Cardano's method:
y
0
1
=
457
51
52
;
y
0
2
=
177
63
382
;
y
0
3
=
235
16
367
;
9
Roots
P
=
y
0
1
=
457
51
52
=
21
83
207
;
Q
=
y
0
3
=
235
16
367
=
15
79
239
;
R
= -
q
8
PQ
=
-34936
1
8
8 *
21
83
207
*
15
79
239
=
13
17
54
;
S
=
bOrigin
4
aOrigin
=
-89
4 *
1
=
-22
1
4
;
λ
0
1
=
P
+
Q
+
R
-
S
=
21
83
207
+
15
79
239
+
13
17
54
-
-22
1
4
=
72
56
191
;
λ
0
2
=
P
-
Q
-
R
-
S
=
21
83
207
-
15
79
239
-
13
17
54
-
-22
1
4
=
15
29
3179
;
λ
0
3
= -
P
+
Q
-
R
-
S
= -
21
83
207
+
15
79
239
-
13
17
54
-
-22
1
4
=
2
97
111
;
λ
0
4
= -
P
-
Q
+
R
-
S
= -
21
83
207
-
15
79
239
+
13
17
54
-
-22
1
4
=
-1
71
414
;
Answer
det(A − λ · I) = 0
λ
0
1
=
72
56
191
;
λ
0
2
=
15
29
3179
;
λ
0
3
=
2
97
111
;
λ
0
4
=
-1
71
414
;
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