Jordan Form Preview

Math Linear Algebra • Linear Transformations and Eigenvalues

Written by STEM Calculators Team

Preview the Jordan canonical form \(J\) for small matrices. Computes eigenvalues, Jordan block sizes using \(\dim\ker(A-\lambda I)^k\), builds Jordan chains (generalized eigenvectors), and (when possible) verifies \(A\approx PJP^{-1}\). Includes a Jordan block visual with a Play highlight animation.

Matrix size

Tolerance

Display decimals

Steps

Allow complex Jordan form (\(\mathbb{C}\)) Normalize chain vectors Scale first nonzero component to 1 Show spectrum plot Show \(P\) and \(P^{-1}\)

Matrix \(A\)

A is 2×2

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e.

Results

Characteristic polynomial

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Eigenvalues (algebraic mult.)

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Jordan blocks

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Reconstruction error \(\|A-PJP^{-1}\|_F\)

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Jordan form \(J\)

Diagonal entries are eigenvalues; ones on the superdiagonal mark chain links.

Chain details

For each eigenvalue \(\lambda\), we show nullities \(\nu_k=\dim\ker(A-\lambda I)^k\) and derived block sizes.

Ready

Jordan blocks visual

Shows the matrix \(J\) with highlighted Jordan blocks. Use Play to cycle highlights.

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Enter matrix \(A\), then click “Calculate”.

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Theory

Jordan form in one line

A matrix \(A\in\mathbb{C}^{n\times n}\) is similar to a Jordan matrix \(J\): \[ A = PJP^{-1}, \] where \(J\) is block diagonal and each block has the form \[ J_s(\lambda)= \begin{bmatrix} \lambda & 1 & & 0\\ & \lambda & \ddots & \\ & & \ddots & 1\\ 0 & & & \lambda \end{bmatrix}. \] Diagonalizable matrices are the special case where every block size is \(1\).

Why matrices fail to diagonalize

If an eigenvalue \(\lambda\) repeats, you may not get enough independent eigenvectors. The eigenspace \[ E_\lambda=\ker(A-\lambda I) \] might have dimension smaller than the algebraic multiplicity of \(\lambda\). This is the classic “defective” case. Jordan form describes exactly how far \(A\) is from being diagonal.

Jordan chains and generalized eigenvectors

For a Jordan block of size \(s\) with eigenvalue \(\lambda\), there is a chain of vectors \[ (A-\lambda I)v_1=0,\quad (A-\lambda I)v_2=v_1,\quad \dots,\quad (A-\lambda I)v_s=v_{s-1}. \] The first vector \(v_1\) is an eigenvector, and \(v_2,\dots,v_s\) are generalized eigenvectors. Placing the chain vectors as columns of \(P\) creates the similarity \(A=PJP^{-1}\) (when \(P\) is invertible).

How block sizes are inferred from nullities

Let \(N=A-\lambda I\). Define \[ \nu_k=\dim\ker(N^k). \] For small matrices, the sequence \(\nu_1,\nu_2,\dots\) determines the Jordan block sizes:

\(\nu_1\) equals the number of Jordan blocks for \(\lambda\).
\(\nu_2-\nu_1\) equals the number of blocks of size at least \(2\).
\(\nu_3-\nu_2\) equals the number of blocks of size at least \(3\), and so on.

This calculator computes \(\nu_k\) (up to \(k=n\)) and reconstructs the block structure from these jumps.

Minimal polynomial (university note)

The minimal polynomial is the smallest-degree monic polynomial \(m(x)\) with \(m(A)=0\). For Jordan form, the exponent of \((x-\lambda)\) in \(m(x)\) equals the size of the largest Jordan block for \(\lambda\). This is why block sizes matter for computations like \(A^k\) and \(e^{tA}\).

Why Jordan form is useful: \(e^{tA}\)

For differential equations \(x'(t)=Ax(t)\), the solution uses the matrix exponential \(e^{tA}\). If \(A=PJP^{-1}\), then \[ e^{tA}=P\,e^{tJ}\,P^{-1}, \] and \(e^{tJ}\) is easier because each Jordan block produces polynomial factors times \(e^{\lambda t}\).