Jordan Canonical Form

Any square matrix over the complex numbers is similar to a block-diagonal matrix in which each block is either a single eigenvalue or a so-called Jordan block with ones on the superdiagonal. This representation, the Jordan canonical form, reveals the algebraic and geometric multiplicities of eigenvalues. For a 2×2 matrix $A$, we have two possibilities: either $A$ is diagonalizable and its Jordan form is a diagonal matrix of eigenvalues, or it is defective and takes the form $\left(\begin{array}{cc}\lambda & 1\\ 0& \lambda \end{array}\right)$.

The diagonalizable case occurs when $A$ has two distinct eigenvalues or has a repeated eigenvalue with two linearly independent eigenvectors. The defective case arises when a repeated eigenvalue has only one eigenvector. Determining which case applies involves solving the characteristic polynomial $\det \left(A-\lambda I\right)=0$ and analyzing the resulting eigenvectors.

Computational Steps

Given $A=\left[\begin{array}{cc}a{11}_{}& a{12}_{}\\ a{21}_{}& a{22}_{}\end{array}\right]$, we compute the trace $t=a{11}_{}+a{22}_{}$ and determinant $d=a{11}_{}a{22}_{}-a{12}_{}a{21}_{}$. The characteristic equation is ${\lambda }^2-t\lambda +d=0$. Its discriminant $\Delta =t^2-4d$ decides whether the eigenvalues are equal.

If $\Delta >0$, the eigenvalues are distinct and $A$ is diagonalizable. If $\Delta =0$, the eigenvalues coincide. In this scenario we examine whether $A-\lambda I$ is the zero matrix. If it is, then $A$ is already a multiple of the identity and diagonalizable. Otherwise, it has only one eigenvector, leading to a single Jordan block.

Applications

The Jordan form clarifies the structure of linear differential equations, matrix exponentials, and similarity transformations. By knowing the Jordan block sizes, one can compute powers of matrices and exponential maps more easily. In control theory and quantum mechanics, such insights simplify systems of linear operators. Understanding the decomposition also highlights when perturbations might cause dramatic changes, since defective matrices are sensitive to small adjustments.

Learning how to compute the Jordan form fosters deeper intuition about eigenvalues, eigenvectors, and their geometric multiplicities. Experiment with various matrices to see how repeated eigenvalues influence the result. This tool can serve as a stepping stone to more advanced linear algebra topics.

Extending to Larger Matrices

The same approach applies to higher-dimensional matrices, though the bookkeeping becomes more involved. After finding each eigenvalue and its algebraic multiplicity, you determine how many Jordan blocks of each size appear. Software libraries like NumPy automate this, but the conceptual process mirrors the 2×2 example here.

Having the full Jordan form clarifies the behavior of linear systems, from controllability in engineering to degeneracy in quantum mechanics. Repeated eigenvalues may form chains that reveal subtle dependencies between solutions.

Numerical Considerations

Real-world matrices often contain measurement noise. Slight perturbations can change whether a matrix appears defective, so checking the sensitivity of eigenvalues to small changes is essential. When instability arises, techniques such as the Schur decomposition or singular value analysis can provide more reliable insight than direct Jordan form computation.

Interpreting the Output

The matrix displayed after calculation is similar to the original matrix through a change of basis. You can verify this by computing $J$ and confirming that $PJ{P}^{-1}$ recreates the input matrix for some invertible $P$. Recording the result with the copy button lets you paste it into algebra systems for further exploration.

Worked Example

Take the matrix $\left[\begin{array}{cc}4& 1\\ 0& 4\end{array}\right]$. The trace equals eight and the determinant sixteen, yielding a discriminant of zero. Because the off-diagonal element $1$ is non-zero, the matrix has only one eigenvector. The calculator reports $\left[\begin{array}{cc}4& 1\\ 0& 4\end{array}\right]$ as its own Jordan form—a single Jordan block with eigenvalue four. If we instead input $\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]$, the discriminant is positive and the tool outputs a diagonal matrix, indicating two independent eigenvectors. This step-by-step process demonstrates how the discriminant determines the Jordan form.

Comparison Table

Matrix Type	Eigenvalues	Jordan Form
Distinct eigenvalues	λ₁≠λ₂	Diagonal diag(λ₁,λ₂)
Repeated eigenvalue, two eigenvectors	λ, λ	Diagonal diag(λ,λ)
Repeated eigenvalue, one eigenvector	λ, λ	Jordan block [[λ,1],[0,λ]]

This table summarizes the possible scenarios for 2×2 matrices. The calculator automates these checks, but understanding the categories deepens comprehension of linear transformations.

Limitations and Assumptions

The calculator focuses on 2×2 matrices and assumes entries represent exact real numbers. Numerical rounding may cause borderline cases when the discriminant is extremely close to zero; in such situations, results should be interpreted cautiously. For larger matrices, computing the full Jordan form can be numerically unstable, and software libraries may prefer the Schur decomposition. Additionally, complex eigenvalues are displayed using JavaScript’s floating-point arithmetic, which may introduce small imaginary components in cases where eigenvalues should be real.

Further Study

After exploring Jordan forms, you might compute functions of matrices using our Matrix Square Root Calculator or analyze logarithms of matrices with the Matrix Logarithm Calculator. These tools build on the eigenvalue insights revealed by the Jordan decomposition.

By demystifying the step-by-step logic behind the Jordan form and highlighting numerical caveats, this calculator serves as both a learning aid and a practical tool for quick canonical decompositions.