Jordan Normal Form Calculator
Calculate jordan normal form instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Jordan Normal Form Calculator
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Formula: A = PJP^(-1), where J is the Jordan normal form
Worked example — J = diag(4.464, -2.464) | Matrix is diagonalizable with distinct eigenvalues
Formula
A = PJP^(-1), where J is the Jordan normal form
Where A is the original matrix, J is the Jordan normal form (block-diagonal with Jordan blocks), and P is the invertible matrix of generalized eigenvectors. Each Jordan block has an eigenvalue on the diagonal and ones on the superdiagonal.
Worked Examples
Example 1: Defective Matrix with Jordan Block
Problem:Find the Jordan normal form of A = [[5, 4], [-1, -3]] if it has a repeated eigenvalue.
Solution:Characteristic polynomial: lambda^2 - 2*lambda - 15 + 4 = lambda^2 - 2*lambda - 11 Wait, let me recalculate: det(A - lambda*I) = (5-lambda)(-3-lambda) - (4)(-1) = -15 + 3*lambda - 5*lambda + lambda^2 + 4 = lambda^2 - 2*lambda - 11 Discriminant = 4 + 44 = 48 > 0 lambda1 = (2 + 6.928) / 2 = 4.464 lambda2 = (2 - 6.928) / 2 = -2.464 Distinct eigenvalues: Jordan form is diagonal.
Result:J = diag(4.464, -2.464) | Matrix is diagonalizable with distinct eigenvalues
Example 2: Matrix Requiring a Jordan Block
Problem:Find the Jordan normal form of A = [[3, 1], [0, 3]].
Solution:Characteristic polynomial: (3-lambda)^2 = 0 lambda = 3 (algebraic multiplicity 2) A - 3I = [[0, 1], [0, 0]] This is NOT the zero matrix, so rank(A-3I) = 1 Geometric multiplicity = 2 - 1 = 1 Since geometric < algebraic, need Jordan block. Jordan form: J = [[3, 1], [0, 3]]
Result:J = [[3, 1], [0, 3]] | Non-diagonalizable, requires 2x2 Jordan block
Frequently Asked Questions
What is the Jordan normal form of a matrix?
The Jordan normal form (also called Jordan canonical form) is a special block-diagonal matrix that is similar to the original matrix. Every square matrix over the complex numbers has a Jordan normal form, written as J = P^(-1)AP, where P is an invertible matrix of generalized eigenvectors. The Jordan form consists of Jordan blocks along the diagonal. Each Jordan block corresponds to an eigenvalue and has that eigenvalue on the diagonal, ones on the superdiagonal, and zeros elsewhere. The Jordan form reveals the complete algebraic structure of a linear transformation, including information that eigenvalues alone cannot convey, particularly about the behavior of repeated eigenvalues and defective matrices.
What is a Jordan block?
A Jordan block is a square matrix with a single eigenvalue lambda on the main diagonal, ones on the superdiagonal (the diagonal just above the main diagonal), and zeros everywhere else. A k-by-k Jordan block is written as J_k(lambda). For example, a 3x3 Jordan block for eigenvalue 2 is [[2,1,0],[0,2,1],[0,0,2]]. A 1x1 Jordan block is simply the eigenvalue itself, which is equivalent to a diagonal entry. When a matrix is diagonalizable, its Jordan form consists entirely of 1x1 Jordan blocks (making it a diagonal matrix). Non-diagonalizable matrices require larger Jordan blocks, which encode the deficiency in eigenvectors. The size and number of Jordan blocks determine the matrix exponential and the behavior of the associated dynamical system.
When is a matrix diagonalizable versus requiring Jordan blocks?
A matrix is diagonalizable if and only if for every eigenvalue, the geometric multiplicity (number of linearly independent eigenvectors) equals the algebraic multiplicity (multiplicity as a root of the characteristic polynomial). When these multiplicities differ, the matrix is defective and requires Jordan blocks larger than 1x1. For example, the matrix [[2,1],[0,2]] has eigenvalue 2 with algebraic multiplicity 2 but geometric multiplicity 1 (only one independent eigenvector), so its Jordan form is a single 2x2 Jordan block. In contrast, the identity matrix [[2,0],[0,2]] has the same eigenvalue with geometric multiplicity 2, making it already diagonal. The Jordan form uniquely characterizes this distinction.
How do you find the Jordan normal form of a 2x2 matrix?
For a 2x2 matrix, the process depends on the eigenvalues. If the two eigenvalues are distinct, the Jordan form is diagonal with the eigenvalues on the diagonal. If the eigenvalue is repeated (algebraic multiplicity 2), compute A - lambda*I. If this equals the zero matrix, the original matrix is already lambda*I (a scalar matrix), and the Jordan form is diagonal. If A - lambda*I is nonzero but has rank 1, the geometric multiplicity is 1, and the Jordan form is [[lambda, 1], [0, lambda]]. For complex eigenvalues a +/- bi, the real Jordan form uses 2x2 blocks [[a, b], [-b, a]]. Jordan Normal Form Calculator handles all three cases automatically and identifies which case applies.
What is the difference between algebraic and geometric multiplicity?
Algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial. Geometric multiplicity is the dimension of the eigenspace, meaning the number of linearly independent eigenvectors for that eigenvalue. The geometric multiplicity is always less than or equal to the algebraic multiplicity. When they are equal for all eigenvalues, the matrix is diagonalizable. The gap between them determines the structure of the Jordan blocks. Specifically, the number of Jordan blocks for an eigenvalue equals its geometric multiplicity, while the sum of all block sizes equals its algebraic multiplicity. Understanding this relationship is critical for advanced linear algebra and applications in differential equations.
How is the Jordan form used in solving systems of differential equations?
The Jordan form simplifies solving systems of linear ordinary differential equations of the form dx/dt = Ax. If A = PJP^(-1), then the substitution y = P^(-1)x transforms the system into dy/dt = Jy, which is much easier to solve because J is block-diagonal. For a diagonal Jordan form, each equation decouples: y_i(t) = c_i * e^(lambda_i * t). For a 2x2 Jordan block with eigenvalue lambda, the solution involves both e^(lambda*t) and t*e^(lambda*t), reflecting the polynomial growth that occurs with defective eigenvalues. This polynomial-times-exponential behavior explains why repeated eigenvalues with Jordan blocks can cause resonance phenomena in physical systems like coupled oscillators.
What is the relationship between Jordan form and matrix powers?
The Jordan form greatly simplifies computing matrix powers. Since A = PJP^(-1), we have A^n = PJ^nP^(-1). Powers of Jordan blocks follow a binomial-like pattern: the (i,j) entry of J_k(lambda)^n is C(n, j-i) * lambda^(n-j+i) when j >= i and 0 otherwise, where C(n,k) is the binomial coefficient. For a diagonal matrix, A^n simply raises each diagonal entry to the n-th power. For a 2x2 Jordan block, J^n = [[lambda^n, n*lambda^(n-1)], [0, lambda^n]]. This formula is essential in computing matrix exponentials, solving recurrence relations, analyzing Markov chains, and studying the long-term behavior of discrete dynamical systems.
How does the Jordan form relate to the minimal polynomial?
The Jordan form directly determines the minimal polynomial of a matrix. The minimal polynomial is the polynomial of lowest degree that, when applied to the matrix, gives the zero matrix. Each distinct eigenvalue contributes a factor (lambda - lambda_i)^k_i, where k_i is the size of the largest Jordan block for that eigenvalue. For example, if the Jordan form has blocks J_2(3) and J_1(3) and J_1(5), the minimal polynomial is (x-3)^2 * (x-5). In contrast, the characteristic polynomial would be (x-3)^3 * (x-5). The Cayley-Hamilton theorem states that every matrix satisfies its characteristic polynomial, but the minimal polynomial provides a tighter algebraic relationship and more information about the matrix structure.
Is the Jordan normal form unique?
The Jordan normal form is unique up to the ordering of Jordan blocks along the diagonal. This means that while you can rearrange the blocks in any order, the sizes and eigenvalues of the blocks are completely determined by the matrix. This uniqueness makes the Jordan form a canonical form, meaning it serves as a standard representative for the similarity class of a matrix. Two matrices are similar (A = PBP^(-1) for some invertible P) if and only if they have the same Jordan normal form. This property makes the Jordan form an invaluable theoretical tool for classifying matrices and proving theorems about linear transformations. The transformation matrix P, however, is not unique.
What are generalized eigenvectors and why are they needed?
Generalized eigenvectors extend the concept of eigenvectors to handle defective matrices. A generalized eigenvector of rank k for eigenvalue lambda satisfies (A - lambda*I)^k * v = 0 but (A - lambda*I)^(k-1) * v is not zero. Ordinary eigenvectors are generalized eigenvectors of rank 1. For a 2x2 Jordan block, we need one ordinary eigenvector v1 and one generalized eigenvector v2 satisfying (A - lambda*I)*v2 = v1. These generalized eigenvectors form chains that correspond to the Jordan blocks. Together, the ordinary and generalized eigenvectors span the entire space, forming the columns of the transformation matrix P. Without generalized eigenvectors, we could not construct a complete basis for defective matrices.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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