If analyzing matrices gives you a headache, this eigenvalue calculator 2x2 is a perfect tool for you. It will allow you to find the trace, determinant, eigenvalues, and eigenvectors of an arbitrary 2x2 matrix. In this article, we will provide you with explanations and handy formulas to ensure you understand how this calculator works. Let's dive right in!
A 2x2 matrix q has the following form:
where a, b, c and d are the elements of the matrix. Our calculator uses the form above, so make sure to input the numbers properly - don't mix them up!
Calculating the trace and determinant
Once you have created a 2x2 matrix, our eigenvalue calculator 2x2 will find the trace and the determinant automatically. If you want to check the correctness, or simply perform the calculations by hand, follow the steps below.
- Trace: the trace of a matrix is defined as the sum of elements on the main diagonal (from upper left to lower right). It is also equal to the sum of the eigenvalues. In the case of our 2x2 matrix,
T = a + d
- Determinant: the determinant of a matrix is useful in multiple further operations - for example while finding the inverse of a matrix. For a 2x2 matrix,
D = ad - bc
Determining the eigenvalues
Each 2x2 matrix q has two eigenvalues: λ₁ and λ₂. These are defined as real numbers that fulfill the following condition for a nonzero column vector v = (v₁, v₂), called an eigenvector:
q * v = λ * v
You can also find another, equivalent version of the equation above:
(q - λI)v = 0
where I is a 2x2 identity matrix.
Knowing the trace and determinant of the matrix q, finding the eigenvalues is a trivial task - all you have to do is input these values into the following equations:
λ₁ = T/2 + √(T²/4 - D)
λ₂ = T/2 - √(T²/4 - D)
Some matrices have only one eigenvalue. Some examples include matrices of the formor
Make sure to experiment with our calculator to see which matrices have only one eigenvalue!
Eigenvector calculator - 2x2 matrices
Our eigenvalue calculator for 2x2 matrices can also be used to find the eigenvectors. It determines these vectors by directly solving the equation
(q - λI)v = 0
Again, a matrix can only have one eigenvector (if it has only one eigenvalue).
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 to the unit vector calculator.