How this instrument works
Diagonalizing a matrix means finding its eigenvalues — special scaling factors along which the matrix acts like plain multiplication by a single number, rather than a full rotation-and-stretch of the whole plane. For a 2×2 matrix, both eigenvalues come from one quadratic equation built entirely from two easy-to-find quantities: the trace (the sum of the diagonal entries) and the determinant (already familiar from this site's other matrix pages).
That quadratic, λ²−(trace)λ+(determinant)=0, is solved with the identical quadratic formula this site's Factoring Trinomials and other algebra pages already rely on — eigenvalues aren't a separate technique so much as the quadratic formula applied to a specific, matrix-derived equation.
Two quick checks confirm a pair of eigenvalues is correct without redoing the whole calculation: their SUM should always equal the trace, and their PRODUCT should always equal the determinant — both must hold simultaneously for genuine eigenvalues.
- Enter the matrix's four entries into the a, b, c, and d fields.
- Read Eigenvalue 1 and Eigenvalue 2 — the two roots of the matrix's own characteristic quadratic.
- Add the two eigenvalues together and confirm the total matches the matrix's trace (a+d).
- Multiply the two eigenvalues together and confirm the product matches the matrix's determinant (ad−bc).
Worked example — [[4,1],[2,3]]
The matrix [[4,1],[2,3]] has a trace of 4+3=7 and a determinant of 4×3−1×2=10, so its eigenvalues solve λ²−7λ+10=0, giving λ=5 and λ=2 via the quadratic formula. Checking: their sum, 5+2=7, matches the trace, and their product, 5×2=10, matches the determinant, exactly as it always must.
The identity matrix [[1,0],[0,1]] has both eigenvalues equal to 1 — it scales every direction by exactly 1, meaning it doesn't transform anything at all. A diagonal matrix like [[2,0],[0,5]] has its eigenvalues sitting right on its own diagonal, 2 and 5, with no calculation needed beyond reading them off directly.
Questions
What is an eigenvalue?
A special scaling factor for a matrix — a number λ such that the matrix, applied to a specific direction (its matching eigenvector), simply scales that direction by λ rather than rotating or reshaping it into something else entirely.
How do you find the eigenvalues of a 2×2 matrix?
Solve the quadratic λ²−(trace)λ+(determinant)=0 using the ordinary quadratic formula — the trace is the sum of the matrix's diagonal entries, and the determinant is the same quantity this site's other matrix pages already compute.
What if the matrix has no real eigenvalues?
That happens when the characteristic quadratic's discriminant, trace²−4·det, comes out negative — the eigenvalues then exist only as a complex-conjugate pair, outside what this page covers.
How can I check my eigenvalues are correct?
Their sum should equal the matrix's trace, and their product should equal its determinant — both relationships must hold simultaneously for a genuine pair of eigenvalues.
What are the eigenvalues of a diagonal matrix?
Simply its own diagonal entries — no calculation is needed at all, since a diagonal matrix already scales each axis independently by exactly the value sitting on that row's diagonal.