Concept

The spectral decomposition of a real symmetric matrix expresses it as $A=Q\Lambda Q^T$. Here $Q$ is orthogonal with columns of unit eigenvectors, and $\Lambda$ is diagonal with eigenvalues. This decomposition diagonalizes the quadratic form represented by $A$, revealing its principal axes.

Diagonalization has far-reaching applications, from solving systems of differential equations to analyzing covariance matrices in statistics. By orthogonally diagonalizing $A$, we can easily raise it to powers or compute functions like $A^{\mathrm{n}}$ by applying the function to the eigenvalues.

For a 2x2 symmetric matrix $\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$, the eigenvalues satisfy the characteristic polynomial ${\lambda }^2-(a+c)\lambda +ac-b^2=0$. The solutions are $\lambda =\frac{a+c\pm \sqrt{a^2-2ac+c^2+4b^2}}{2}$.

The eigenvectors are obtained by solving $(A-\lambda I)v=0$ for each eigenvalue. For example, when $\lambda 1_{}$ is known, the vector $(v1,v2)$ satisfying $(a-\lambda 1_{})v1+bv2=0$ may be chosen. Normalizing gives the columns of $Q$.

This calculator automates the algebra for 2x2 symmetric matrices to provide immediate insight into how eigenvalues and eigenvectors work together. Try experimenting with different values to see how the matrix orientation and scaling change.