Eigenvalue Calculator NZ
This calculator finds the eigenvalues of a 2 by 2 matrix, the special numbers that reveal how the matrix stretches and transforms space. Eigenvalues are among the most important concepts in linear algebra, with deep applications across science and engineering: they describe the natural frequencies of vibrating systems, the stability of structures and control systems, the principal directions in statistics and machine learning through principal component analysis, and the behaviour of quantum systems. An eigenvalue of a matrix is a scaling factor for a special direction, an eigenvector, that the matrix stretches without rotating. For a 2 by 2 matrix there are two eigenvalues, found by solving the characteristic equation, which conveniently reduces to a quadratic built from the matrix's trace and determinant. This tool computes them. You enter the four entries of the matrix, and the calculator returns the two eigenvalues, along with the trace, the sum of the diagonal, the determinant, and the discriminant that decides whether the eigenvalues are real or complex. When the discriminant is negative the eigenvalues are a complex conjugate pair, which the calculator reports in the form real plus or minus imaginary i. The results update as you type. Use it for linear algebra homework, for analysing the stability or dynamics of a system, or as a step toward finding eigenvectors. Two useful checks the calculator highlights: the two eigenvalues always add up to the trace, and they always multiply to give the determinant, which is a quick way to verify the result. Real, distinct eigenvalues mean the matrix simply stretches along two directions, while complex eigenvalues signal rotation, as in an oscillating or spiralling system.
Eigenvalues solve x² - (trace)x + det = 0. They sum to the trace and multiply to the determinant. A negative discriminant gives a complex conjugate pair.
How it works
For a 2 by 2 matrix the eigenvalues are the roots of the characteristic equation, lambda squared minus the trace times lambda plus the determinant equals zero. Using the quadratic formula, they are the trace plus or minus the square root of the discriminant, all over two, where the discriminant is the trace squared minus four times the determinant.
Worked example
For the matrix with rows (2, 1) and (1, 2), the trace is 4 and the determinant is 3. The discriminant is 4 squared minus 4 times 3, which is 4. The eigenvalues are (4 plus or minus the square root of 4) over 2, giving 3 and 1. As a check, they sum to 4, the trace, and multiply to 3, the determinant.
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