Educational tool; results are estimates for learning, verify with your textbook or instructor.
This tool finds eigenvalues of a single $2\times 2$ real matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Eigenvalues are the roots of the characteristic polynomial $\lambda^2-(\mathrm{tr}\,A)\lambda+\det A=0$, where $\mathrm{tr}\,A=a+d$ and $\det A=ad-bc$.
The roots are meaningful when you want the stretch factors along directions that $A$ does not tip. A real eigenvalue $\lambda$ means some nonzero vector $\mathbf{v}$ satisfies $A\mathbf{v}=\lambda\mathbf{v}$. A complex conjugate pair means the real plane map rotates and scales, with no real invariant line.
This is not a full eigensolver. It does not return eigenvectors, and it does not prove diagonalizability. It solves a quadratic. For a 3 by 3 matrix, the characteristic polynomial and a real eigenvector are on the 3x3 eigenvalue calculator. For the geometry, read the eigenvalue explainer.
How to use the characteristic polynomial
Enter the four entries $a,b,c,d$. The default $\begin{pmatrix}2&1\\1&2\end{pmatrix}$ has trace $4$ and determinant $3$, so the polynomial is $\lambda^2-4\lambda+3=0$ and the roots are $3$ and $1$. Press Find eigenvalues and confirm those two numbers.
The page reports the trace, the determinant, the discriminant of the quadratic, and the roots. Use that audit line. If you only stare at the $\lambda$ list, you miss the arithmetic that produced it.
How to read the output
Two distinct real roots mean two real stretch factors. A repeated real root means a double algebraic eigenvalue; the geometry may or may not have two independent eigenvectors, and this page will not decide that.
A negative discriminant produces $\alpha\pm\beta i$. Those are still eigenvalues over $\mathbb{C}$. Over $\mathbb{R}$ you should picture a rotation scaling rather than two real invariant lines.
The sum of the eigenvalues equals the trace. The product equals the determinant. If a reported pair fails those two checks, something is off. On a healthy run they should match.
Common pitfalls
A repeated root is numerically fragile. A discriminant that should be zero may print as a tiny positive or negative number, splitting one eigenvalue into two close values or into a tiny imaginary part. If you expected a perfect square, check the entries.
Eigenvalues are not entries of $A$ except in special cases (triangular matrices, for example). Do not read $a$ and $d$ as the spectrum unless you know the matrix is triangular.
Without eigenvectors you cannot yet diagonalize, and you cannot yet say which direction stretches. The number $\lambda$ is the factor. The vector is the direction. This page only has the factor.
Related reading
The geometry is in eigenvalues in plain language and in the eigenvalue, eigenvector, and eigenspace entries. The surrounding map is linear algebra. A study order is the linear algebra first course pathway.
Build the determinant and products first on the 2 by 2 matrix calculator. Local linearization in several variables is Jacobian matrices in plain language. The same quadratic algebra appears on the quadratic formula calculator.
Elimination and a larger inverse are different jobs. The reduced row echelon form calculator writes each elementary step of a linear system. The 3 by 3 matrix calculator expands a determinant and builds an inverse when one exists. Neither of those pages finds eigenvalues. The 3x3 eigenvalue calculator writes the characteristic polynomial and row reduces $A-\lambda I$.
