Eigenvalue Calculator
This eigenvalue calculator is an online tool designed to find the eigenvalues and eigenvectors of a given matrix quickly and accurately. It works specifically with 2×2 and 3×3 square matrices, making it perfect for students and educators in linear algebra who need to solve small matrices without tedious manual calculations. With this calculator, you can input your matrix and obtain its eigenvalues (the special scalar values), the corresponding eigenvectors, and even the characteristic polynomial of the matrix in seconds.
Eigenvalue Calculator
Find Eigenvalues and Eigenvectors Step-by-Step
About Eigenvalues and Eigenvectors
For a square matrix A, eigenvalue λ and eigenvector v satisfy: Av = λv. This calculator finds all eigenvalues by solving the characteristic equation det(A - λI) = 0, then computes corresponding eigenvectors.
Eigenvalues and eigenvectors are fundamental in linear algebra with applications in physics, engineering, data analysis, and machine learning. This calculator supports 2×2 and 3×3 matrices with exact solutions.
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Table of Contents
How to Use the Calculator
1.Select the matrix size from the size dropdown (choose either 2×2 or 3×3). The calculator will adjust the input grid accordingly.
2.Enter the values of your matrix into the input grid. Fill in each entry cell with a number (you can use integers, decimals, or negative values as needed). For example, a 2×2 matrix will have 4 input fields (two for the first row and two for the second row), whereas a 3×3 matrix will have 9 fields (three per row).
3.Once your matrix is entered, click the Calculate button. The calculator will compute the results instantly.
4.View the output results on the page. The calculator will display the eigenvalues, each with its corresponding eigenvector, and the characteristic polynomial of the matrix. These results will typically appear below or next to the input grid for easy reference.
5.If you want to analyze another matrix, simply change the matrix size or edit the entries and hit Calculate again. You can experiment with different matrices to see how their eigenvalues and eigenvectors differ.
Pro Tip: Try starting with a simple example, such as a diagonal matrix (all off-diagonal entries zero). For a diagonal matrix, the eigenvalues are just the diagonal entries, and the eigenvectors align with the coordinate axes (the standard basis vectors). This can help you verify the calculator’s output and build intuition before moving on to more complex matrices.
Matrix Size (2×2 vs 3×3)
Begin by choosing the size of the matrix you want to work with. The calculator supports square matrices of size 2×2 or 3×3. A 2×2 matrix has 2 rows and 2 columns, while a 3×3 matrix has 3 rows and 3 columns. You can switch between these options using a size selector in the interface. Only square matrices are accepted because eigenvalues are defined for square matrices (the number of rows must equal the number of columns).
Matrix Entries
After selecting the matrix size, you’ll see an input grid with blank fields corresponding to each entry of the matrix. Fill each field with the numerical value for that position in the matrix. You can use any real numbers for the entries – positive, negative, or zero – and decimals or whole numbers are all fine (for example, use 0.5 for one-half). If an entry is left blank, the calculator may treat it as 0 by default, so it’s best to fill in all fields intentionally. The first row of input fields corresponds to the first row of the matrix, the second row of fields to the second matrix row, and so on. Double-check that you’ve entered the numbers in the correct positions (for example, don’t accidentally swap a row with a column). Once all entries are in place, you’re ready to calculate the eigenvalues and eigenvectors for that matrix.
Results Explained
After you hit Calculate, the tool will display three main results for your matrix:
the eigenvalues,
the eigenvectors (one for each eigenvalue),
and the characteristic polynomial.
To understand these, let’s break down what each term means and how to interpret the values. We’ll use a simple example matrix to illustrate each result.
Eigenvalues
Eigenvalues are the special scalar values that describe how a matrix scales certain vectors. If you input a matrix $A$, its eigenvalues $\lambda$ are the solutions to the equation $\det(A – \lambda I) = 0$ (where $I$ is the identity matrix; this is called the characteristic equation). In simpler terms, an eigenvalue is a number such that when the matrix multiplies its corresponding eigenvector, the result is just the eigenvector scaled by that number.
For example, consider the matrix $A = \begin{pmatrix}2 & 1\\[6pt] 1 & 2\end{pmatrix}$. The calculator would find two eigenvalues for $A$. In this case, the eigenvalues are $\lambda_1 = 3$ and $\lambda_2 = 1$. These values mean that there are certain special vectors that $A$ stretches by a factor of 3 or by a factor of 1 (i.e. leaves unchanged in magnitude). If an eigenvalue is positive, it indicates the eigenvector is stretched in the same direction; if it’s negative, the eigenvector is flipped in direction (because multiplying by a negative scalar reverses the vector’s direction). An eigenvalue of 0 would mean the matrix squashes some vector completely to the zero vector (this also tells you the matrix is singular, or non-invertible).
In our example, $\lambda_1 = 3$ being an eigenvalue tells us there’s a special direction in which matrix $A$ triples any vector lying in that direction. $\lambda_2 = 1$ as an eigenvalue indicates there’s another direction that $A$ leaves unchanged in length (since multiplying by 1 doesn’t change magnitude). The calculator lists these eigenvalues usually in a set or list. If a matrix had a repeated eigenvalue (for instance, two eigenvalues both equal to 5), the tool would list that value with its multiplicity or simply show the same value twice.
Eigenvectors
Eigenvectors are the special vectors associated with each eigenvalue. An eigenvector is a nonzero vector $\mathbf{v}$ that satisfies $A\mathbf{v} = \lambda \mathbf{v}$, meaning that when the matrix $A$ multiplies $\mathbf{v}$, the result is just $\mathbf{v}$ scaled by the eigenvalue $\lambda$. In other words, the direction of $\mathbf{v}$ doesn’t change under the transformation represented by $A$ – it’s only stretched or shrunk.
For our example matrix $A = \begin{pmatrix}2 & 1\\[3pt] 1 & 2\end{pmatrix}$, the calculator will provide an eigenvector for each eigenvalue. Let’s say it outputs:
For $\lambda_1 = 3$, an eigenvector $\mathbf{v}_1 = \begin{pmatrix}1\\[3pt]1\end{pmatrix}$.
For $\lambda_2 = 1$, an eigenvector $\mathbf{v}_2 = \begin{pmatrix}1\\[3pt]-1\end{pmatrix}$.
These are example eigenvectors that correspond to the eigenvalues 3 and 1, respectively. You can verify this is correct by actually multiplying the matrix by those vectors:
\[
A \begin{pmatrix} 1 \\ 1 \end{pmatrix}
= \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}
\begin{pmatrix} 1 \\ 1 \end{pmatrix}
= \begin{pmatrix} 3 \\ 3 \end{pmatrix}
= 3 \begin{pmatrix} 1 \\ 1 \end{pmatrix},
\]
which shows that \(\begin{pmatrix}1 \\ 1\end{pmatrix}\) is scaled by 3 (matching \(\lambda_1 = 3\)).
Likewise,
\[
A \begin{pmatrix} 1 \\ -1 \end{pmatrix}
= \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}
\begin{pmatrix} 1 \\ -1 \end{pmatrix}
= \begin{pmatrix} 1 \\ -1 \end{pmatrix}
= 1 \begin{pmatrix} 1 \\ -1 \end{pmatrix},
\]
confirming \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) is an eigenvector with eigenvalue \(1\).
Each eigenvalue-eigenvector pair tells us that the matrix transforms that eigenvector by simply stretching/compressing or flipping it. Important: Eigenvectors are not unique in length or sign – if $\mathbf{v}$ is an eigenvector, then any scaled version of $\mathbf{v}$ (like $2\mathbf{v}$ or $-1\mathbf{v}$) is also an eigenvector of the same eigenvalue. The calculator typically provides one convenient representative for each eigenvalue (often normalized or with simple integers). You can always multiply an eigenvector by any nonzero constant and it remains a valid eigenvector.
Characteristic Polynomial
The characteristic polynomial of a matrix is a polynomial expression that encapsulates the eigenvalues. It is defined by the determinant $\det(A – \lambda I)$ and usually written as $p(\lambda)$. When you set this polynomial equal to zero, you get the characteristic equation $\det(A – \lambda I)=0$, whose solutions are exactly the eigenvalues of $A$.
For a 2×2 matrix, the characteristic polynomial is a quadratic (degree 2) polynomial. In our example with $A = \begin{pmatrix}2 & 1\ 1 & 2\end{pmatrix}$, the calculator outputs the characteristic polynomial:
\[
p(\lambda) = \lambda^2 – 4\lambda + 3.
\]
This polynomial comes from computing $\det(A – \lambda I)$ for the matrix $A$. If we factor this polynomial, we get:
\[
\lambda^2 – 4\lambda + 3 = (\lambda – 1)(\lambda – 3),
\]
which clearly has roots $\lambda = 1$ and $\lambda = 3$. Those roots are exactly the eigenvalues we found earlier ($\lambda_1=1$ and $\lambda_2=3$). The characteristic polynomial thus provides a unified way to represent all eigenvalues: it’s zero precisely at the eigenvalues.
For a 3×3 matrix, the characteristic polynomial will be a cubic (degree 3) polynomial. It might be more complicated to factor, and sometimes it has one or more complex roots. Regardless, the calculator will compute the polynomial for you. By examining the coefficients of the characteristic polynomial, you can also glean information: for instance, the constant term of the polynomial (up to sign) is the determinant of $A$, and the coefficient of the $\lambda^2$ (for a 3×3 case) or $\lambda$ (for a 2×2 case) term is related to the trace of $A$ (the sum of the diagonal entries), which is also the sum of the eigenvalues. These relationships can help you double-check results. But if you’re just getting started, the key takeaway is that the characteristic polynomial is another representation of the eigenvalues, and the calculator provides it to help you see how the eigenvalues come from solving $p(\lambda)=0$.
Eigenvalue and Eigenvector Formulas
The eigenvalues of a matrix are found by solving the characteristic equation:
\[
\det(A – \lambda I) = 0,
\]
where $I$ is the identity matrix. Expanding this determinant yields a polynomial in $\lambda$ (the characteristic polynomial). Solving that polynomial equation gives the eigenvalues $\lambda$s.
2×2 Case: If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the characteristic polynomial will be a quadratic:
\[
\det \begin{pmatrix} a – \lambda & b \\ c & d – \lambda \end{pmatrix} = 0,
\]
which expands to:
\[
\lambda^2 – (a + d)\lambda + (ad – bc) = 0.
\]
Here $a+d$ is the trace of $A$ (sum of diagonal entries) and $ad – bc$ is the determinant of $A$. Using the quadratic formula, the two eigenvalues are:
\[
\lambda_{1,2} = \frac{(a+d) \pm \sqrt{(a+d)^2 – 4(ad – bc)}}{2}.
\]
For example, for $A = \begin{pmatrix}2 & 1 \\ 1 & 2\end{pmatrix}$, we have $a+d = 2+2 = 4$ and $ad – bc = (2)(2) – (1)(1) = 3$. Plugging these into the formula:
\[
\lambda_{1,2} = \frac{4 \pm \sqrt{4^2 – 4(3)}}{2} = \frac{4 \pm \sqrt{16 – 12}}{2} = \frac{4 \pm 2}{2}.
\]
This gives $\lambda_1 = 3$ and $\lambda_2 = 1$, matching what we found earlier.
3×3 Case: For a 3×3 matrix, the characteristic polynomial is cubic (degree 3). Finding its roots can be challenging (there’s no simple formula for a general cubic), so one usually factors the polynomial if possible or uses a computational tool. Our calculator takes care of solving the cubic equation for you, so you don’t need to do it by hand.
Eigenvector Formula: Once you have an eigenvalue $\lambda$, you find its eigenvector by solving the linear system
\[
(A – \lambda I)\,x = 0,
\]
where $\mathbf{0}$ is the zero vector. This system of equations will give the relationships between components of the eigenvector. For example, for $\lambda = 3$ in our $2\times2$ example,
\[
A – 3I = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} – \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix},
\]
and solving $(A-3I)\,\mathbf{x}=0$ yields the condition $-x_1 + x_2 = 0$ (from the first row; the second row gives the same equation), so $x_2 = x_1$. Thus we can choose $x_1 = 1$, giving an eigenvector $\mathbf{v}_1 = (1,1)^T$.
Similarly, for $\lambda = 1$,
\[
A – I = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} – \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix},
\]
so $(A-I)\,\mathbf{x}=0$ leads to $x_1 + x_2 = 0$, giving an eigenvector like $(1,-1)^T$. In practice, you can scale the eigenvector differently (e.g., $c(2,-2)^T$ is also fine for the same eigenvalue) — any nonzero scalar multiple is an acceptable eigenvector. The calculator will typically give you one convenient choice for each eigenvalue.
FAQ - Eigenvalue Calculator
What is an eigenvalue?
An eigenvalue is a number (scalar) that tells you how much a matrix stretches or shrinks an eigenvector. Formally, if a vector $\mathbf{v}$ is an eigenvector of a matrix $A$, and $A\mathbf{v} = \lambda \mathbf{v}$, then the scalar $\lambda$ is the eigenvalue. In simple terms, it’s the factor by which the matrix multiplies (scales) that special vector $\mathbf{v}$. Each eigenvalue corresponds to at least one eigenvector direction that the matrix transforms by that factor.
What is an eigenvector?
An eigenvector is a special vector that only gets scaled (stretched or compressed) when a matrix is applied to it, not rotated to a new direction. In other words, if $\mathbf{v}$ is an eigenvector of matrix $A$ with eigenvalue $\lambda$, then $A\mathbf{v} = \lambda \mathbf{v}$. The vector $\mathbf{v}$ keeps its direction (or exactly opposite direction if $\lambda$ is negative) after the transformation; only its length changes by the factor $\lambda$. Eigenvectors are by definition nonzero and are unique up to scaling (any multiple of an eigenvector is also an eigenvector in the same direction).
How do you find eigenvalues of a 2×2 or 3×3 matrix?
To find eigenvalues of a 2×2 or 3×3 matrix, you set up and solve the characteristic equation: $\det(A – \lambda I) = 0$. This involves computing the determinant of the matrix $A$ minus $\lambda$ times the identity matrix. For a 2×2 matrix, this yields a quadratic equation in $\lambda$; you can solve it using the quadratic formula to get two eigenvalues. For a 3×3 matrix, you get a cubic equation; you may factor it (if possible) or use a calculator to find the three eigenvalues. Our eigenvalue calculator automates this process, finding the roots of the characteristic polynomial and giving you the eigenvalues instantly.
What is the characteristic polynomial of a matrix?
The characteristic polynomial of a matrix is the polynomial you get from $\det(A – \lambda I)$. It is a function of $\lambda$ (often denoted $p(\lambda)$) that encodes the matrix’s eigenvalues. If you set the characteristic polynomial equal to zero ($p(\lambda)=0$), you’re writing the characteristic equation whose solutions are the eigenvalues. For example, if a matrix’s characteristic polynomial is $p(\lambda) = \lambda^2 – 5\lambda + 6$, then solving $p(\lambda)=0$ (i.e. $\lambda^2 – 5\lambda + 6 = 0$) will give eigenvalues $\lambda = 2$ and $\lambda = 3$.
What are eigenvalues and eigenvectors used for?
Eigenvalues and eigenvectors are fundamental in many areas of science and engineering. They are used to understand the behavior of linear transformations and systems. For example, in physics, eigenvalues can represent natural vibration frequencies of a system (with eigenvectors as vibration modes). In computer science and data analysis, techniques like Principal Component Analysis (PCA) use eigenvectors and eigenvalues to reduce data dimensions and find important features. In graph theory, eigenvalues of a network’s adjacency matrix can reveal properties of the network. Essentially, eigenvalues and eigenvectors help simplify complex matrix operations by finding special directions (eigenvectors) where the transformation acts like simple scaling by a factor (the eigenvalue).
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