# Chapter 68. Jordan Canonical Form # Chapter 68. Jordan Canonical Form Jordan canonical form describes the structure of a square matrix that may fail to be diagonalizable. Diagonalization is the ideal case. If a matrix has enough independent eigenvectors, it can be written as $$ A=PDP^{-1}, $$ where \(D\) is diagonal. But many matrices do not have enough eigenvectors. Jordan canonical form gives the next best representation. It replaces a defective matrix by a block diagonal matrix that is almost diagonal. Over an algebraically closed field, such as \(\mathbb{C}\), every square matrix is similar to a matrix built from Jordan blocks. Each block has one eigenvalue on the diagonal and ones immediately above the diagonal. This form records eigenvalues, algebraic multiplicities, geometric multiplicities, and the failure of diagonalization. ## 68.1 The Problem After Diagonalization A matrix \(A\) is diagonalizable if there is a basis of eigenvectors. In that case, the matrix of the transformation in that basis is diagonal. But consider $$ A= \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}. $$ The only eigenvalue is $$ \lambda=2. $$ Compute $$ A-2I= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. $$ Solving $$ (A-2I)v=0 $$ gives $$ y=0. $$ Thus $$ E_2= \operatorname{span} \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}. $$ There is only one independent eigenvector, but the space is two-dimensional. Hence \(A\) cannot be diagonalized. Still, \(A\) has a simple structure. It is already a Jordan block: $$ \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}. $$ It acts like multiplication by \(2\), together with a nilpotent shift. ## 68.2 Jordan Blocks A Jordan block of size \(k\) with eigenvalue \(\lambda\) is the \(k\times k\) matrix $$ J_k(\lambda)= \begin{bmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ 0 & 0 & \lambda & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & 1 \\ 0 & 0 & 0 & 0 & \lambda \end{bmatrix}. $$ It has \(\lambda\) on the main diagonal, \(1\) on the superdiagonal, and \(0\) elsewhere. For example, $$ J_3(5)= \begin{bmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{bmatrix}. $$ A Jordan block of size \(1\) is just $$ J_1(\lambda)= [\lambda]. $$ Thus diagonal matrices are special cases of Jordan forms in which every Jordan block has size \(1\). ## 68.3 Jordan Canonical Form A Jordan canonical form is a block diagonal matrix whose diagonal blocks are Jordan blocks: $$ J= \begin{bmatrix} J_{k_1}(\lambda_1) & 0 & \cdots & 0 \\ 0 & J_{k_2}(\lambda_2) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & J_{k_r}(\lambda_r) \end{bmatrix}. $$ A square matrix \(A\) has Jordan canonical form \(J\) if there exists an invertible matrix \(P\) such that $$ A=PJP^{-1}. $$ Equivalently, $$ P^{-1}AP=J. $$ The columns of \(P\) form a special basis called a Jordan basis. The Jordan form is unique up to the order of its blocks. The order of blocks may be changed, but the block sizes attached to each eigenvalue are determined by the matrix. ## 68.4 A Jordan Block as Scalar Plus Nilpotent A Jordan block can be written as $$ J_k(\lambda)=\lambda I+N, $$ where $$ N= \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ 0 & 0 & 0 & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}. $$ The matrix \(N\) is nilpotent. This means that some power of \(N\) is zero. For a \(k\times k\) Jordan block, $$ N^k=0, $$ but $$ N^{k-1}\neq 0. $$ Thus a Jordan block consists of two parts: | Part | Meaning | |---|---| | \(\lambda I\) | diagonal scaling by the eigenvalue | | \(N\) | nilpotent shift along a chain | The diagonal part describes eigenvalue scaling. The nilpotent part describes the failure of diagonalization. ## 68.5 Generalized Eigenvectors To build Jordan form, ordinary eigenvectors are not enough. We need generalized eigenvectors. A nonzero vector \(v\) is a generalized eigenvector of \(A\) for eigenvalue \(\lambda\) if $$ (A-\lambda I)^m v=0 $$ for some positive integer \(m\). Every ordinary eigenvector is a generalized eigenvector with \(m=1\), since $$ (A-\lambda I)v=0. $$ Generalized eigenvectors allow us to complete missing eigenvector bases. For the defective matrix $$ A= \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}, $$ there is only one ordinary eigenvector. But there are two generalized eigenvectors, enough to form a basis. ## 68.6 Jordan Chains A Jordan chain for eigenvalue \(\lambda\) is a list of vectors $$ v_1,v_2,\ldots,v_k $$ such that $$ (A-\lambda I)v_1=0 $$ and $$ (A-\lambda I)v_2=v_1, $$ $$ (A-\lambda I)v_3=v_2, $$ and so on, until $$ (A-\lambda I)v_k=v_{k-1}. $$ The first vector \(v_1\) is an ordinary eigenvector. The later vectors are generalized eigenvectors. Equivalently, $$ Av_1=\lambda v_1, $$ $$ Av_2=\lambda v_2+v_1, $$ $$ Av_3=\lambda v_3+v_2, $$ and in general, $$ Av_j=\lambda v_j+v_{j-1}. $$ This chain produces one Jordan block of size \(k\). ## 68.7 Matrix of a Jordan Chain Let $$ v_1,v_2,\ldots,v_k $$ be a Jordan chain for eigenvalue \(\lambda\). In the ordered basis $$ v_1,v_2,\ldots,v_k, $$ the action of \(A\) is $$ Av_1=\lambda v_1, $$ $$ Av_2=v_1+\lambda v_2, $$ $$ Av_3=v_2+\lambda v_3, $$ and so on. Thus the matrix has the form $$ \begin{bmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ 0 & 0 & \lambda & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & 1 \\ 0 & 0 & 0 & 0 & \lambda \end{bmatrix}. $$ This is exactly the Jordan block \(J_k(\lambda)\). Thus Jordan blocks are the coordinate matrices of Jordan chains. ## 68.8 Example: A Two by Two Jordan Block Let $$ A= \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}. $$ The eigenvalue is \(\lambda=2\). Let $$ N=A-2I= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. $$ We have $$ Ne_1=0 $$ and $$ Ne_2=e_1. $$ Therefore $$ e_1,e_2 $$ is a Jordan chain. The matrix \(A\) in this basis is already $$ J_2(2)= \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}. $$ The vector \(e_1\) is an eigenvector. The vector \(e_2\) is a generalized eigenvector. ## 68.9 Example: A Larger Jordan Block Consider $$ J_3(\lambda)= \begin{bmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{bmatrix}. $$ Let $$ N=J_3(\lambda)-\lambda I = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}. $$ Then $$ Ne_1=0, $$ $$ Ne_2=e_1, $$ $$ Ne_3=e_2. $$ Also, $$ N^2e_3=e_1 $$ and $$ N^3e_3=0. $$ Thus \(e_3\) is a generalized eigenvector of rank \(3\), and $$ e_1,e_2,e_3 $$ form a Jordan chain. ## 68.10 Algebraic and Geometric Multiplicity in Jordan Form Jordan form makes algebraic and geometric multiplicities visible. For a fixed eigenvalue \(\lambda\): | Quantity | Jordan form interpretation | |---|---| | Algebraic multiplicity | Total size of all Jordan blocks for \(\lambda\) | | Geometric multiplicity | Number of Jordan blocks for \(\lambda\) | | Largest Jordan block size | Exponent of \((t-\lambda)\) in the minimal polynomial | For example, $$ J= \begin{bmatrix} 4 & 1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 4 & 1 & 0 \\ 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 4 \end{bmatrix} $$ has three Jordan blocks for eigenvalue \(4\): two blocks of size \(2\) and one block of size \(1\). Therefore: $$ \text{algebraic multiplicity}=5, $$ $$ \text{geometric multiplicity}=3. $$ ## 68.11 Diagonalization as a Special Case A matrix is diagonalizable exactly when every Jordan block has size \(1\). If all Jordan blocks are \(1\times 1\), then the Jordan form is diagonal: $$ J= \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix}. $$ If any Jordan block has size greater than \(1\), the matrix is not diagonalizable. Thus Jordan canonical form extends diagonalization to defective matrices. ## 68.12 Kernels and Block Sizes The dimensions of kernels $$ \ker(A-\lambda I), \quad \ker(A-\lambda I)^2, \quad \ker(A-\lambda I)^3, \quad \ldots $$ determine the Jordan block sizes for \(\lambda\). Let $$ N=A-\lambda I $$ on the generalized eigenspace for \(\lambda\). Then $$ \dim \ker N $$ is the number of Jordan blocks. The difference $$ \dim \ker N^2-\dim \ker N $$ counts how many blocks have size at least \(2\). More generally, $$ \dim \ker N^r-\dim \ker N^{r-1} $$ counts how many blocks have size at least \(r\). This gives a systematic way to recover Jordan block sizes. ## 68.13 Example from Kernel Dimensions Suppose for an eigenvalue \(\lambda\), the generalized eigenspace has dimension \(6\), and $$ \dim \ker N=2, $$ $$ \dim \ker N^2=4, $$ $$ \dim \ker N^3=5, $$ $$ \dim \ker N^4=6. $$ Then: $$ \dim \ker N-\dim \ker N^0=2 $$ so there are \(2\) blocks of size at least \(1\). $$ \dim \ker N^2-\dim \ker N=2 $$ so there are \(2\) blocks of size at least \(2\). $$ \dim \ker N^3-\dim \ker N^2=1 $$ so there is \(1\) block of size at least \(3\). $$ \dim \ker N^4-\dim \ker N^3=1 $$ so there is \(1\) block of size at least \(4\). Thus the block sizes are $$ 4 \qquad \text{and} \qquad 2. $$ The Jordan form for this eigenvalue has blocks $$ J_4(\lambda) \oplus J_2(\lambda). $$ ## 68.14 Generalized Eigenspaces The generalized eigenspace for eigenvalue \(\lambda\) is $$ G_\lambda = \ker(A-\lambda I)^m $$ for \(m\) large enough. If the algebraic multiplicity of \(\lambda\) is \(a\), it is enough to take $$ m=a. $$ Thus $$ G_\lambda=\ker(A-\lambda I)^a. $$ The generalized eigenspace contains the ordinary eigenspace: $$ E_\lambda=\ker(A-\lambda I)\subseteq G_\lambda. $$ The space decomposes as a direct sum of generalized eigenspaces: $$ V= G_{\lambda_1}\oplus G_{\lambda_2}\oplus\cdots\oplus G_{\lambda_k}. $$ On each generalized eigenspace, the operator has only one eigenvalue and can be studied through a nilpotent part. ## 68.15 Jordan Decomposition on a Generalized Eigenspace On \(G_\lambda\), define $$ N=A-\lambda I. $$ Then \(N\) is nilpotent on \(G_\lambda\). This means $$ N^m=0 $$ for some \(m\). Therefore, on \(G_\lambda\), $$ A=\lambda I+N. $$ The Jordan form of \(A\) on \(G_\lambda\) is determined by the Jordan form of the nilpotent operator \(N\). This reduces the study of Jordan form to the study of nilpotent operators. ## 68.16 Nilpotent Jordan Form A nilpotent matrix has only one eigenvalue: $$ 0. $$ Its Jordan form consists of blocks $$ J_k(0)= \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ 0 & 0 & 0 & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}. $$ Thus every Jordan block for a general eigenvalue is just a nilpotent Jordan block shifted by \(\lambda I\): $$ J_k(\lambda)=\lambda I+J_k(0). $$ Nilpotent behavior measures how far a matrix is from being diagonalizable. ## 68.17 Powers of a Jordan Block Let $$ J=\lambda I+N, $$ where \(N^k=0\). Then for a positive integer \(m\), $$ J^m=(\lambda I+N)^m. $$ Since \(\lambda I\) commutes with \(N\), the binomial theorem applies: $$ J^m = \sum_{r=0}^{k-1} \binom{m}{r} \lambda^{m-r}N^r. $$ Terms with \(r\geq k\) vanish because $$ N^k=0. $$ This formula shows that powers of a Jordan block contain both exponential behavior from \(\lambda^m\) and polynomial factors from the nilpotent part. ## 68.18 Matrix Functions Jordan form also explains matrix functions. If $$ J=\lambda I+N $$ and \(N^k=0\), then for a polynomial or analytic function \(f\), $$ f(J) = f(\lambda I+N). $$ Using Taylor expansion, $$ f(J)= f(\lambda)I + f'(\lambda)N + \frac{f''(\lambda)}{2!}N^2 + \cdots + \frac{f^{(k-1)}(\lambda)}{(k-1)!}N^{k-1}. $$ Thus a function of a Jordan block depends not only on \(f(\lambda)\), but also on derivatives of \(f\) at \(\lambda\). This is one reason Jordan form is theoretically powerful. ## 68.19 Jordan Form and Differential Equations Consider the linear differential equation $$ x'(t)=Ax(t). $$ Its solution is $$ x(t)=e^{tA}x(0). $$ If $$ A=PJP^{-1}, $$ then $$ e^{tA}=Pe^{tJ}P^{-1}. $$ For a Jordan block $$ J=\lambda I+N, $$ we have $$ e^{tJ} = e^{t\lambda}e^{tN}. $$ Since \(N\) is nilpotent, $$ e^{tN} = I+tN+\frac{t^2}{2!}N^2+\cdots+\frac{t^{k-1}}{(k-1)!}N^{k-1}. $$ Therefore Jordan blocks produce solutions involving $$ e^{\lambda t}, \quad t e^{\lambda t}, \quad t^2 e^{\lambda t}, \quad \ldots $$ This explains why repeated eigenvalues in differential equations can produce polynomial factors. ## 68.20 Jordan Form and Minimal Polynomial The minimal polynomial of \(A\) is the monic polynomial of least degree \(m_A(t)\) such that $$ m_A(A)=0. $$ Jordan form makes it easy to read. If the largest Jordan block for eigenvalue \(\lambda\) has size \(s_\lambda\), then the minimal polynomial contains the factor $$ (t-\lambda)^{s_\lambda}. $$ Thus $$ m_A(t) = \prod_{\lambda} (t-\lambda)^{s_\lambda}. $$ The characteristic polynomial records the total size of all blocks. The minimal polynomial records the largest block size for each eigenvalue. A matrix is diagonalizable exactly when the minimal polynomial has no repeated root. ## 68.21 Existence of Jordan Form Over \(\mathbb{C}\), every square matrix has a Jordan canonical form. More generally, a matrix has a Jordan canonical form over a field \(F\) if its characteristic polynomial splits completely over \(F\). That means all eigenvalues lie in \(F\). If the characteristic polynomial does not split, the matrix may not have Jordan form over that field. For example, the real matrix $$ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$ has characteristic polynomial $$ t^2+1. $$ It has no real eigenvalues, so it has no real Jordan form with real eigenvalues. Over \(\mathbb{C}\), it has eigenvalues \(i\) and \(-i\), and it is diagonalizable. ## 68.22 Numerical Warning Jordan canonical form is mainly a theoretical tool. It is highly sensitive to small perturbations. A matrix with a repeated eigenvalue and a Jordan block may become diagonalizable after an arbitrarily small perturbation. Conversely, nearly repeated eigenvalues can make numerical Jordan computations unstable. For numerical work, one usually prefers Schur decomposition, singular value decomposition, or other stable factorizations. Jordan form classifies exact algebraic structure, but it is usually avoided as a computational normal form in floating point arithmetic. ## 68.23 Common Errors The first common error is to think that every repeated eigenvalue creates a Jordan block larger than \(1\). A repeated eigenvalue may still have enough eigenvectors. The second common error is to confuse algebraic multiplicity with block size. Algebraic multiplicity is the total size of all blocks for an eigenvalue, not the size of one block. The third common error is to forget generalized eigenvectors. Jordan bases are built from chains, not only from ordinary eigenvectors. The fourth common error is to assume Jordan form is numerically safe. It is structurally exact but computationally unstable. The fifth common error is to ignore the field. Jordan form with scalar eigenvalues requires the characteristic polynomial to split over the field being used. ## 68.24 Summary Jordan canonical form expresses a square matrix as $$ A=PJP^{-1}, $$ where \(J\) is block diagonal and each block has the form $$ J_k(\lambda)= \begin{bmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ 0 & 0 & \lambda & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & 1 \\ 0 & 0 & 0 & 0 & \lambda \end{bmatrix}. $$ A Jordan block represents one chain of generalized eigenvectors. Diagonalization is the special case in which every Jordan block has size \(1\). Jordan form records the precise failure of diagonalization. It shows how eigenvalues, eigenspaces, generalized eigenvectors, nilpotent parts, minimal polynomials, and matrix functions fit into one structure.