Jordan Normal Form Calculator

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Jordan Normal Form Calculator

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Eigenvalue and generalized-vector reasoning
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What Is Jordan Normal Form?

Jordan normal form, also called Jordan canonical form, is a structured way to represent a square matrix after a change of basis. It is most useful when a matrix is close to being diagonalizable but does not have enough independent eigenvectors to become a diagonal matrix.

A diagonal matrix is the simplest possible form: each coordinate direction is scaled independently. Jordan normal form is the next best thing. It keeps the eigenvalues on the diagonal, but it may include \(1\) entries just above the diagonal. Those \(1\) entries show where ordinary eigenvectors must be extended into generalized eigenvector chains.

A Jordan block for an eigenvalue \(\lambda\) has the form

$$ J_k(\lambda)= \begin{bmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & \lambda & 1 \\ 0 & \cdots & \cdots & 0 & \lambda \end{bmatrix}. $$

The subscript \(k\) is the size of the block. A size-\(1\) block is simply the \(1\times1\) matrix \([\lambda]\), so a diagonal matrix is a Jordan matrix whose blocks are all size \(1\).

When a matrix \(A\) can be written as

$$ A=PJP^{-1}, $$

\(J\) is the Jordan normal form and \(P\) is the change-of-basis matrix. Equivalently,

$$ J=P^{-1}AP. $$

The two equations say the same thing: \(A\) and \(J\) represent the same linear transformation in different bases.

Why Jordan Normal Form Matters

Jordan normal form explains exactly how diagonalization can fail. If a matrix has enough independent eigenvectors, its Jordan form is diagonal. If it does not, the larger Jordan blocks show the missing eigenvectors and how generalized eigenvectors fill the gap.

This matters in linear algebra because many matrix problems become easier after changing basis. Diagonalization simplifies powers, recurrences, differential equations, and the study of a linear transformation's long-term behavior. Jordan normal form extends that idea to matrices with repeated eigenvalues that are not diagonalizable.

For students, Jordan normal form is also a useful diagnostic tool. It separates three related ideas that are often confused:

  • the eigenvalues of a matrix,
  • the multiplicity of each eigenvalue,
  • the number of independent eigenvectors available for each eigenvalue.

Key Terms to Know

  • Eigenvalue: A scalar \(\lambda\) such that \(Av=\lambda v\) for some nonzero vector \(v\).
  • Eigenvector: A nonzero vector \(v\) whose direction is preserved by the matrix transformation \(A\).
  • Eigenspace: The null space \(\ker(A-\lambda I)\), containing all eigenvectors for \(\lambda\) together with the zero vector.
  • Algebraic multiplicity: How many times \(\lambda\) appears as a root of the characteristic polynomial.
  • Geometric multiplicity: The dimension of the eigenspace \(\ker(A-\lambda I)\).
  • Nullity: The dimension of the null space of a matrix.
  • Generalized eigenvector: A nonzero vector \(w\) such that \((A-\lambda I)^k w=0\) for some positive integer \(k\), even if \(w\) is not an ordinary eigenvector.
  • Jordan chain: A linked list of vectors satisfying relations such as \((A-\lambda I)w=v\), where \(v\) is an eigenvector.
  • Similarity: Two matrices are similar if they represent the same linear transformation in different bases, usually written as \(B=P^{-1}AP\).

How Jordan Normal Form Works

The process starts with eigenvalues. For a square matrix \(A\), the characteristic polynomial is

$$ p(\lambda)=\det(A-\lambda I). $$

The roots of this polynomial are the eigenvalues. For a \(2\times2\) matrix

$$ A=\begin{bmatrix}a&b\\c&d\end{bmatrix}, $$

its trace and determinant are

$$ \operatorname{tr}(A)=a+d, \qquad \det(A)=ad-bc. $$

The characteristic polynomial is

$$ p(\lambda)=\lambda^2-\operatorname{tr}(A)\lambda+\det(A), $$

so the eigenvalues are

$$ \lambda=\frac{\operatorname{tr}(A)\pm\sqrt{\operatorname{tr}(A)^2-4\det(A)}}{2}. $$

For an upper- or lower-triangular matrix, the eigenvalues are the diagonal entries. This is why triangular matrices are especially convenient for small Jordan form calculations.

After the eigenvalues are known, the main question is how many Jordan blocks are attached to each eigenvalue and how large those blocks are. For an eigenvalue \(\lambda\), set

$$ N_\lambda=A-\lambda I. $$

The geometric multiplicity is

$$ m_g(\lambda)=\dim\ker(A-\lambda I). $$

This number is also the number of Jordan blocks associated with \(\lambda\). The algebraic multiplicity \(m_a(\lambda)\) is the total size of all those blocks.

The matrix is diagonalizable exactly when every eigenvalue has enough eigenvectors:

$$ m_g(\lambda)=m_a(\lambda) $$

for every eigenvalue \(\lambda\). Equivalently, the total number of independent eigenvectors equals the matrix size \(n\):

$$ \sum_i m_g(\lambda_i)=n. $$

When the geometric multiplicity is smaller than the algebraic multiplicity, at least one Jordan block must have size greater than \(1\).

How Nullities Reveal Block Sizes

Nullities of powers of \(A-\lambda I\) reveal the detailed block pattern. Let

$$ d_k=\dim\ker\left((A-\lambda I)^k\right), \qquad d_0=0. $$

Then

$$ d_k-d_{k-1} $$

is the number of Jordan blocks for \(\lambda\) whose size is at least \(k\).

For example, if the nullities for an eigenvalue are

$$ d_1=1,\qquad d_2=2,\qquad d_3=3, $$

then the differences are

$$ d_1-d_0=1, \qquad d_2-d_1=1, \qquad d_3-d_2=1. $$

There is one block of size at least \(1\), one block of size at least \(2\), and one block of size at least \(3\). Therefore, the Jordan structure for that eigenvalue is one size-\(3\) block.

Examples of Jordan Normal Form in Practice

Example 1: A \(2\times2\) Matrix That Is Not Diagonalizable

Consider

$$ A=\begin{bmatrix}5&1\\0&5\end{bmatrix}. $$

This matrix is already a Jordan block. Its only eigenvalue is \(\lambda=5\), with algebraic multiplicity \(2\):

$$ p(\lambda)=(\lambda-5)^2. $$

Now compute

$$ A-5I= \begin{bmatrix}0&1\\0&0\end{bmatrix}. $$

The null space is spanned by

$$ \begin{bmatrix}1\\0\end{bmatrix}, $$

so the geometric multiplicity is \(1\). Since \(m_a(5)=2\) but \(m_g(5)=1\), the matrix is not diagonalizable.

There is one Jordan block, and its total size must be \(2\):

$$ J=\begin{bmatrix}5&1\\0&5\end{bmatrix}. $$

The generalized eigenvector chain is visible from

$$ (A-5I)\begin{bmatrix}0\\1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}. $$

The vector on the right is an eigenvector, and the vector on the left is a generalized eigenvector.

Example 2: A \(2\times2\) Matrix That Is Diagonalizable

Consider

$$ A=\begin{bmatrix}2&0\\0&5\end{bmatrix}. $$

The eigenvalues are \(2\) and \(5\). They are distinct, so each has algebraic multiplicity \(1\) and geometric multiplicity \(1\). The matrix has two independent eigenvectors, so it is diagonalizable.

Its Jordan form is diagonal:

$$ J=\begin{bmatrix}2&0\\0&5\end{bmatrix}. $$

The block pattern is one size-\(1\) block for \(2\) and one size-\(1\) block for \(5\).

Example 3: A Triangular \(3\times3\) Matrix With One Jordan Chain

Consider

$$ A=\begin{bmatrix} 4&1&0\\ 0&4&1\\ 0&0&4 \end{bmatrix}. $$

Because the matrix is triangular, its only eigenvalue is the diagonal entry \(4\), repeated three times. Thus \(m_a(4)=3\).

Let

$$ N=A-4I= \begin{bmatrix} 0&1&0\\ 0&0&1\\ 0&0&0 \end{bmatrix}. $$

The nullities are

$$ \dim\ker(N)=1, \qquad \dim\ker(N^2)=2, \qquad \dim\ker(N^3)=3. $$

The differences are all \(1\), so there is one block of size at least \(1\), one block of size at least \(2\), and one block of size at least \(3\). Therefore, the Jordan form has a single size-\(3\) block:

$$ J=\begin{bmatrix} 4&1&0\\ 0&4&1\\ 0&0&4 \end{bmatrix}. $$

This example shows why a repeated eigenvalue is not enough information by itself. The nullities determine whether the repeated eigenvalue splits into several small blocks or one larger block.

How to Interpret the Result

A result of Diagonalizable means the matrix has enough independent eigenvectors for its size. In Jordan form, this means every block is size \(1\), so \(J\) is diagonal.

A result of Not diagonalizable means at least one eigenvalue has geometric multiplicity smaller than algebraic multiplicity. In Jordan form, this creates at least one block of size greater than \(1\).

The Jordan block summary uses a pattern such as

$$ \lambda: 2+1. $$

This means the eigenvalue \(\lambda\) has two Jordan blocks: one block of size \(2\) and one block of size \(1\). The total size, \(2+1=3\), is the algebraic multiplicity of that eigenvalue.

The eigenvalue table separates algebraic multiplicity from geometric multiplicity. This is the key comparison:

  • If \(m_a=m_g\), that eigenvalue contributes only size-\(1\) blocks.
  • If \(m_g<m_a\), that eigenvalue contributes at least one larger block.

When \(P\) and \(J\) are shown, they describe the similarity decomposition

$$ A=PJP^{-1}. $$

The columns of \(P\) are built from eigenvectors and, when needed, generalized eigenvectors. The Jordan block diagram shows the same structure visually: eigenvalues appear on the diagonal, and \(1\) entries above the diagonal show linked generalized eigenvectors within a block.

A Complex eigenvalues result for a real \(2\times2\) matrix means the real-number mode has reached a case outside its supported Jordan form output. It does not mean the matrix has no eigenvalues over the complex numbers.

Common Mistakes and Misconceptions

Assuming a repeated eigenvalue means “not diagonalizable.” A repeated eigenvalue can still be diagonalizable. For example, \(A=5I_2\) has the eigenvalue \(5\) with algebraic multiplicity \(2\) and geometric multiplicity \(2\), so it is diagonalizable.

Confusing algebraic multiplicity with geometric multiplicity. Algebraic multiplicity comes from the characteristic polynomial. Geometric multiplicity comes from the dimension of an eigenspace. Jordan block structure depends on both.

Looking only at the eigenvalues. Eigenvalues tell you what appears on the diagonal of \(J\), but they do not by themselves determine the sizes of the Jordan blocks. Nullities of \(A-\lambda I\) and its powers provide the missing information.

Entering a general \(3\times3\) matrix when only triangular \(3\times3\) matrices are supported. A triangular matrix has eigenvalues on the diagonal, which makes the small-matrix workflow manageable. A general non-triangular \(3\times3\) matrix may require additional characteristic-polynomial and eigenvector calculations that are outside this calculator mode.

Expecting parameters or typed radical expressions. Decimal and scientific-notation entries are converted to exact rational numbers, and real quadratic eigenvalues are returned with exact radicals. Inputs such as parameters, sqrt(2), and typed fractions such as 1/3 are not accepted; enter a decimal or scientific-notation number instead.

Typing expressions instead of numeric entries. Matrix entries should be real numbers. Inputs such as symbolic radicals, parameters, or text fractions may not be accepted as numeric entries.

Ignoring complex eigenvalues. Some real matrices have nonreal eigenvalues. Those matrices can be studied over the complex numbers, but the supported real \(2\times2\) workflow stops rather than producing a complex Jordan decomposition.

When to Use Jordan Normal Form

Use Jordan normal form when you want to understand the structure of a square matrix beyond its eigenvalues alone.

It is especially useful when:

  • a matrix has repeated eigenvalues,
  • you need to decide whether a matrix is diagonalizable,
  • an eigenspace has smaller dimension than expected,
  • you want to identify generalized eigenvector chains,
  • you are studying matrix powers, matrix exponentials, systems of differential equations, or recurrence relations,
  • you need a compact summary of how a linear transformation behaves after a change of basis.

For quick classroom or homework checks, small \(2\times2\) matrices and triangular \(3\times3\) matrices are good settings because the eigenvalues and nullities can be interpreted directly.

Limitations and Things to Keep in Mind

Jordan normal form is an exact algebraic concept. This calculator parses every accepted decimal or scientific-notation entry as an exact rational number, so zero tests, eigenvalue grouping, row reduction, nullities, and Jordan block decisions do not use a floating-point tolerance.

This calculator is designed for real numeric matrices of size \(2\times2\) or \(3\times3\). For \(3\times3\) matrices, the supported case is triangular matrices: entries above or below the diagonal may be nonzero, but not both sides at once.

The main limitations are:

  • It does not support \(1\times1\) matrices, general \(n\times n\) matrices, or matrices larger than \(3\times3\).
  • It does not support general non-triangular \(3\times3\) matrices.
  • It does not support complex-valued matrix entries.
  • It does not compute a real canonical form for real matrices with complex eigenvalues.
  • It does not accept symbolic expressions such as parameters or radicals.
  • Each entry is limited to 64 source characters, scientific exponents from -100 to 100, and an expanded reduced numerator and denominator of at most 128 digits.
  • A full \(P\) and \(J\) decomposition may not be constructed for every supported non-diagonalizable case, even when the Jordan block structure is reported.

Exact mode displays reduced fractions and exact quadratic radicals. Decimal mode changes only the presentation and may round for readability. A \(3\times3\) matrix is triangular only when the required off-diagonal entries are exactly zero. Whenever \(P\) and \(J\) are displayed, the calculator first verifies exactly that \(P\) is invertible and \(AP=PJ\), and it also reports a normalized decimal residual as a diagnostic.

For proofs, exams, publications, or any work where exact algebra matters, use the calculator as a guide and verify the result by hand or with an exact symbolic algebra system.

How to Use This Calculator

  1. Choose the matrix size: \(2\times2\) or \(3\times3\).
  2. Enter each real numeric entry of \(A\) in the matrix grid.
  3. For a \(3\times3\) matrix, make sure the matrix is triangular.
  4. Alternatively, paste a square \(2\times2\) or \(3\times3\) matrix using spaces, commas, tabs, semicolons, or new lines as separators.
  5. Select the output format mode to switch between exact fractions/radicals and decimal approximations.
  6. Review the main result to see whether the matrix is diagonalizable, not diagonalizable, unsupported, or has complex eigenvalues.
  7. Check the eigenvalue table to compare algebraic multiplicity, geometric multiplicity, and block sizes.
  8. Review the \(J\) and \(P\) matrices when they are provided.
  9. Use the Jordan block diagram to visualize the block pattern, and download it as a PNG if needed.

Frequently Asked Questions

Is a matrix with a repeated eigenvalue always non-diagonalizable?

No. A repeated eigenvalue only means the algebraic multiplicity is greater than \(1\). The matrix is still diagonalizable if the eigenspace has the same dimension as that algebraic multiplicity.

What does a Jordan block of size \(2\) mean?

A size-\(2\) Jordan block means there is one ordinary eigenvector and one generalized eigenvector linked to it. The superdiagonal \(1\) in the block records that link.

Why do nullities of powers of \(A-\lambda I\) matter?

The nullity of \(A-\lambda I\) gives the number of Jordan blocks for \(\lambda\). The way the nullity grows for powers such as \((A-\lambda I)^2\) and \((A-\lambda I)^3\) reveals how long those blocks are.

What does \(\lambda: 2+1\) mean in a block summary?

It means the eigenvalue \(\lambda\) has one Jordan block of size \(2\) and one Jordan block of size \(1\). The total algebraic multiplicity for that eigenvalue is \(3\).

Why might a \(3\times3\) matrix be rejected?

The supported \(3\times3\) mode is for triangular matrices. If a matrix has nonzero entries both above and below the diagonal, it is a general \(3\times3\) matrix and is outside this calculator mode.

What exactly does exact mode guarantee?

Accepted decimal and scientific-notation entries are exact rationals internally. Exact mode shows reduced fractions and exact quadratic radicals for supported real \(2\times2\) eigenvalues. It does not accept symbolic parameters or radical expressions as input.

What happens if a \(2\times2\) matrix has complex eigenvalues?

The calculator reports complex eigenvalues as outside the supported real Jordan form mode. Over the complex numbers, the matrix can still have complex eigenvalues and may have a complex Jordan form, but that is not the same as the supported real-number output.

Sources and References

Books and Open Textbooks

  1. Sheldon Axler. Linear Algebra Done Right. 4th ed., Springer, 2024. Chapters 8A–8C on null spaces of powers, generalized eigenvectors, generalized eigenspace decomposition, and Jordan form. https://linear.axler.net/LADR4e.pdf
  2. Dan Margalit and Joseph Rabinoff. Interactive Linear Algebra. Georgia Institute of Technology. Sections 5.2 and 5.4 on the characteristic polynomial, triangular matrices, diagonalization, algebraic multiplicity, and geometric multiplicity. https://textbooks.math.gatech.edu/ila/

Online and University Sources

  1. Russell Herman. “2.10: Appendix—Diagonalization and Linear Systems.” A Second Course in Ordinary Differential Equations: Dynamical Systems and Boundary Value Problems, Mathematics LibreTexts, accessed June 28, 2026. https://math.libretexts.org/Bookshelves/DifferentialEquations/ASecondCourseinOrdinaryDifferentialEquations%3ADynamicalSystemsandBoundaryValueProblems%28Herman%29/02%3ASystemsofDifferentialEquations/2.10%3AAppendix-DiagonalizationandLinear_Systems
  2. Georgia Institute of Technology. “Jordan Canonical Forms.” December 6, 2006. Used for the relationship between nullities of powers of \(A-\lambda I\) and Jordan block sizes. https://ecroot.math.gatech.edu/jordan.pdf