# Eigenvalue and Eigenvector Calculator

Created by Bogna Szyk and Maciej Kowalski, PhD candidate
Reviewed by Jack Bowater and Steven Wooding
Last updated: Jan 18, 2024

If analyzing matrices gives you a headache, this eigenvalue and eigenvector calculator is the perfect tool for you. It will allow you to find the eigenvalues of a matrix of size 2x2 or 3x3 matrix and will even save you time by finding the eigenvectors as well. In this article, we will provide you with explanations and handy formulas to ensure you understand how this calculator works and how to find eigenvalues and eigenvectors in general.

Let's dive right in!

## 2x2 matrix

A 2x2 matrix $A$ has the following form:

$A = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \end{bmatrix}$

where $a_1$, $a_2$, $b_1$ and $b_2$ are the elements of the matrix. Our eigenvalue and eigenvector calculator uses the form above, so make sure to input the numbers properly – don't mix them up!

## Calculating the trace and determinant

In the case of a 2x2 matrix, in order to find the eigenvectors and eigenvalues, it's helpful first to get two very special numbers: the trace and the determinant of the array. Lucky for us, the eigenvalue and eigenvector calculator will find them automatically, and if you'd like to see them, click on the advanced mode button. In case you want to check if it gave you the correct answer or simply perform the calculations by hand, follow the steps below:

1. Trace: the trace of a matrix is defined as the sum of the elements on the main diagonal (from the upper left to the lower right). It is also equal to the sum of the eigenvalues (counted with their multiplicities). In the case of a 2x2 matrix, it is:
$\qquad \mathrm{tr}(A) = a_1+b_1$
1. Determinant: the determinant of a matrix is useful in multiple further operations – for example, finding the inverse of a matrix (you can learn how to do it at our inverse matric calculator). For a 2x2 matrix, the determinant is:
$\qquad |A| = a_1b_2 - a_2b_1$

## How to find eigenvalues

Each 2x2 matrix $A$ has two eigenvalues: $\lambda_1$ and $\lambda_2$. These are defined as numbers that fulfill the following condition for a nonzero column vector $\bold{v} = (v_1, v_2)$, which we call an eigenvector:

$A \times v = \lambda \times v$

You can also find another equivalent version of the equation above:

$(A -\lambda \mathbb{I})v = 0$

where $\mathbb{I}$ is the 2x2 identity matrix.

Knowing the trace and determinant, it is a trivial task to find the eigenvalues of a matrix – all you have to do is input these values into the following equations:

$\lambda_1= \frac{\mathrm{tr}(A)}{2} + \sqrt{\frac{\mathrm{tr}(A)^2}{4} - |A|}$

And:

$\lambda_2= \frac{\mathrm{tr}(A)}{2} - \sqrt{\frac{\mathrm{tr}(A)^2}{4} - |A|}$

Some matrices have only one eigenvalue. Examples of such arrays include matrices of the form:

$A = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$

or...

$A = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$

Make sure to experiment with our calculator to see which matrices have only one eigenvalue!

## Eigenvalue and eigenvector calculator – 2x2 matrices

You can also use our calculator for finding eigenvectors. In essence, learning how to find eigenvectors boils down to directly solving the equation:

$(q-\lambda\mathbb{I})v=0$

Note that if a matrix has only one eigenvalue, it can still have multiple eigenvectors corresponding to it. For instance, the identity matrix:

$\mathbb{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

...has only one (double) eigenvalue $\lambda = 1$, but two eigenvectors: $v_1 = (1,0)$ and $v_2 = (0,1)$.

Remember that if a vector $v$ is an eigenvector, then the same vector multiplied by a scalar is also an eigenvector of the same matrix. If you would like to simplify the solution provided by our calculator, head over to the unit vector calculator.

## How to find eigenvalues and eigenvectors of 3x3 matrices

Let's now try to translate all this into the language of 3x3 matrices. First of all, let's see an example of such an object:

$A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}$

where for us, the entries $a_1$, $a_2$, up to $c_3$ are real numbers.

In general, most of the definitions above are the same for 3x3 matrices. For instance, the trace is the sum of the cells on the main diagonal:

$\mathrm{tr}(A) = a_! + b_2 + c_3$

However, the determinant is now a more complicated manner:

$\begin{split} |A| &= a_1\!\times \!b_2\!\times \!c_3 + a_2\!\times \!b_3 \!\times \!c_1\\ &+ a_3\!\times \!b_1 \!\times \!c_2 - a_3\!\times \!b_2 \!\times \!c_1 \\ &- a_2\!\times \!b_1\!\times \!c_3 - a_1\!\times \!b_3\!\times \!c_2 \end{split}$

Now, when it comes to how to find eigenvectors and eigenvalues, the definition is again the same: they are the numbers $\lambda$ and vectors $v$ that satisfy the matrix equation:

$A \times v = \lambda\times v$

where the multiplication on the left is matrix multiplication (a rather complex operations we detailed at our matrix multiplication calculator). However, the trick is that this time the equation is far more complicated. In particular, the formulas from above don't work here.

In the case of 2x2 matrices, it all boils down to the quadratic formula: if you don't remember how to solve it, check out our quadratic formula calculator. However, when the arrays are of size 3x3, we obtain a cubic equation, i.e., an equation with the variable to the third power. And such things are not so easy to calculate.

Fortunately, we have the eigenvalue and eigenvector calculator that can hide all these ugly formulas and effortlessly give us a pretty answer.

But is the answer always a pretty one?

## Complex eigenvalues and eigenvectors

Quadratic and cubic equations sometimes have no real solutions. This means that there is no real number (the kind of number that we learned when we were little kids) that satisfies this formula. Therefore, in the field of real numbers, it's not always possible to find the eigenvalues of a matrix. However, in mathematics, there is an extension in which that can never happen: every equation has as many solutions (counted with their multiplicities) as its degree.

Complex numbers, formally speaking, are pairs of real numbers. The first of the pair is called the real part, and the second the imaginary part (yup, that's exactly what professional mathematicians called it). The second one has the mysterious number $\mathrm i$, which we define as the square root of ($-1$). They told us at school that such things don't exist, didn't they? Well, they do, but they're imaginary.

For us, this means that the calculator will always know how to find the eigenvectors and eigenvalues of a matrix. Once it does that, it's crucial to know if the problem you're solving uses complex numbers or just the real ones. Just to be on the safe side, our eigenvalue and eigenvector calculator will show you all the values and their corresponding eigenvectors, be they real or complex.

However, if you only need the real ones, feel free to ignore all that have an $\mathrm i$ in them. Just keep in mind that they do exist, even though they're imaginary.

## FAQ

### How do I find eigenvalues and eigenvectors?

To find an eigenvalue, λ, and its eigenvector, v, of a square matrix, A, you need to:

1. Write the determinant of the matrix, which is A - λI with I as the identity matrix.

2. Solve the equation det(A - λI) = 0 for λ (these are the eigenvalues).

3. Write the system of equations Av = λv with coordinates of v as the variable.

4. For each λ, solve the system of equations, Av = λv.

5. Write the solution of Av = λv with parameters.

6. For each parameter, its coefficient is the coordinate of an eigenvector.

7. Group the coefficients corresponding to each parameter to form an eigenvector v.

### How do I find eigenvalues of a 3x3 matrix?

To find the eigenvalues λ₁, λ₂, λ₃ of a 3x3 matrix, A, you need to:

1. Subtract λ (as a variable) from the main diagonal of A to get A - λI.
2. Write the determinant of the matrix, which is A - λI.
3. Solve the cubic equation, which is det(A - λI) = 0, for λ.
4. The (at most three) solutions of the equation are the eigenvalues of A.
5. If needed, proceed to find the eigenvectors of the eigenvalues.

### How do I find eigenvectors from eigenvalues?

When you have an eigenvalue, λ, of a square matrix, A, and you want to find its corresponding eigenvector, v, you need to:

1. Denote the coordinates of v as variables (e.g., v = (x,y,z) for 3x3 matrices).
2. Write the system of equations, Av = λv (each coordinate gives one equation).
3. Solve the system of equations for the coordinates of v.
4. Write the solution using parameters.
5. For each parameter, its coefficient is the coordinate of an eigenvector.
6. Group the coefficients corresponding to each parameter to form an eigenvector, v.

### How many eigenvalues does a matrix have?

A square matrix with n rows and columns can have at most n eigenvalues. If we don't allow complex numbers, it may happen that it will have none (i.e., when the characteristic polynomial has no real solutions).

### Are eigenvectors orthogonal?

In general, no. If the initial matrix is symmetric, then the eigenvectors of distinct eigenvalues are always orthogonal.

### Can 0 be an eigenvalue?

Yes, it can. For that to happen, there must exist a non-zero vector, v, such that Av = 0 (as a matrix multiplication).

Bogna Szyk and Maciej Kowalski, PhD candidate
Matrix size
2×2
A=
 ⌈ a1 a2 ⌉ ⌊ b1 b2 ⌋
First row
a₁
a₂
Second row
b₁
b₂
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