Generalized Eigenvector Calculator

Some matrices run out of ordinary eigenvectors: a repeated eigenvalue produces only one independent direction, and diagonalization stops there. Enter a 2×2 or 3×3 matrix and one of its eigenvalues, and this calculator builds the Jordan chain for that eigenvalue: the ordinary eigenvector v1, then the generalized eigenvectors v2 and v3 that solve (A − λI)v2 = v1 and (A − λI)v3 = v2, with a numeric check printed for every link in the chain.

Quick Answer

How to use the generalized eigenvector calculator

You supply the matrix and one eigenvalue; the calculator does the three solves that build the chain. Four steps cover it.

1. Get λ first if you need to. This tool does not hunt for eigenvalues. If you do not already know one, run your matrix through the eigenvalues calculator first and bring one eigenvalue back here. A repeated eigenvalue is the interesting case, since a simple eigenvalue can never start a chain longer than 1.
2. Pick the size and type the matrix. Choose 2×2 or 3×3, then fill the entries row by row. The pre-filled entries are the defective textbook matrix [[2, 1], [0, 2]], so the first click shows a complete chain.
3. Click Calculate and read the verdict badge. The calculator tries a chain from every independent starting eigenvector and reports the longest one it finds. A green badge says every chain stops at v1 for this λ, so the eigenvalue is not defective. An amber badge says the eigenvalue is defective here and reports the longest chain length found.
4. Read down the verification rows. The results box prints (A − λI)v1, then (A − λI)v2 − v1, then (A − λI)v3 − v2 when a v3 exists. Each should be the zero vector to 4 decimals. If a row is not zero, something was typed wrong.

What a generalized eigenvector is, in plain terms

A generalized eigenvector of rank 2 is a vector v2 that (A − λI) sends to an ordinary eigenvector, not to zero. Rank 1 is an ordinary eigenvector; rank 3 is sent to a rank 2 vector, and so on up the chain.

Ordinary eigenvectors are the vectors a matrix simply stretches: Av = λv. A diagonalizable matrix has enough independent eigenvectors to form a basis, so every vector can be written as a mix of pure stretches. Trouble starts with a repeated eigenvalue. The matrix [[2, 1], [0, 2]] has eigenvalue 2 twice, yet solving (A − 2I)v = 0 gives only multiples of [1, 0]. One direction cannot span the plane, so no basis of eigenvectors exists and the matrix cannot be diagonalized. That shortfall is what defective means.

Generalized eigenvectors fill the gap. The missing direction [0, 1] is not an eigenvector, since A[0, 1] = [1, 2], not a multiple of [0, 1]. But (A − 2I)[0, 1] = [1, 0], the eigenvector itself. So [0, 1] sits one step away from the eigenspace, and together the pair {[1, 0], [0, 1]} is a basis again. A set of vectors linked this way, each sent to the previous one by (A − λI) and the first sent to zero, is a Jordan chain. Stack the chains for every eigenvalue as columns of a matrix P and A becomes PJP⁻¹, where J is the Jordan normal form: λ down the diagonal, a 1 above the diagonal for each chain link, zeros elsewhere. As the Wolfram MathWorld page on generalized eigenvectors puts it, these vectors enlarge a too-small set of eigenvectors into a full basis exactly when a matrix fails to be diagonalizable.

The recipe: three solves, in order

The whole method is repeated row reduction of the same shifted matrix. Form M = A − λI once, then solve Mv1 = 0, Mv2 = v1, and (for 3×3) Mv3 = v2, stopping at the first solve that has no solution.

Two details decide whether the arithmetic behaves. First, the solver inside this calculator uses Gaussian elimination to reduced row echelon form with partial pivoting and a 1e-10 tolerance, which is why the zero first row of [[0, 1], [0, 0]] no longer stops it: it swaps and works with free variables instead of back-substituting down a diagonal that has a zero on it. Second, scaling. An eigenvector is only defined up to a multiple, so this page fixes v1 by dividing through until its largest component is 1. The chain vectors after that are not free to be rescaled: change v1 and v2 must change to match, since Mv2 has to equal the v1 you chose. With free variables set to 0, each next vector is then determined.

Worked example 1: the defective 2×2, by hand

For A = [[2, 1], [0, 2]] and λ = 2, the chain is v1 = [1, 0], v2 = [0, 1]. Here is the hand working the calculator repeats.

Step 1, shift. M = A − 2I = [[0, 1], [0, 0]].

Step 2, eigenvector. Mv = 0 reads: 0·x + 1·y = 0, and 0 = 0. So y = 0 and x is free. Taking x = 1 gives v1 = [1, 0], already in this page's scaling, since its largest component is 1. The null space is one-dimensional, so the geometric multiplicity of λ = 2 is 1, against an algebraic multiplicity of 2 from the characteristic polynomial (λ − 2)². Defective, by definition.

Step 3, generalized eigenvector. Solve Mv2 = v1: 0·x + 1·y = 1, and 0 = 0. So y = 1, x free, and setting the free variable to 0 gives v2 = [0, 1].

Step 4, verify. Mv2 = [[0, 1], [0, 0]] × [0, 1] = [1, 0] = v1, and Mv1 = [0, 0]. The chain has length 2, which is the size of the single Jordan block for λ = 2. Type the matrix in above and the results box prints exactly these vectors and checks.

Worked example 2: a chain of length 3

A single 3×3 Jordan block gives the longest possible chain: v1 = [1, 0, 0], v2 = [0, 1, 0], v3 = [0, 0, 1] for λ = 2. Switch the size selector to 3×3 and the entries are already this matrix.

Here A = [[2, 1, 0], [0, 2, 1], [0, 0, 2]], so M = A − 2I = [[0, 1, 0], [0, 0, 1], [0, 0, 0]]. Solving Mv = 0 forces the second and third components to 0, leaving v1 = [1, 0, 0]. Solving Mv2 = v1 gives second component 1 and third 0, so v2 = [0, 1, 0]. Solving Mv3 = v2 gives third component 1, so v3 = [0, 0, 1]. One more solve, Mv4 = v3, would need a row of M to produce the third component of v3 from nothing: it has no solution, and the chain ends at length 3. The table below collects the matrices used on this page and what the tool returns for each.

Algebraic vs geometric multiplicity: the two counts that decide everything

An eigenvalue is defective exactly when its algebraic multiplicity, how many times it appears as a root of the characteristic polynomial, exceeds its geometric multiplicity, the dimension of its eigenspace. Equal counts for every eigenvalue means diagonalizable; any shortfall means a chain is needed.

The geometric multiplicity is the number the calculator can hand you directly: it is the nullity of A − λI, the count of free variables in the first solve, printed in the results box. The set of all ordinary eigenvectors for one λ is its eigenspace, and counting a basis for it is the job of the basis for the eigenspace calculator. When the eigenspaces of all eigenvalues together span the whole space, their bases combine into an eigenbasis and the matrix diagonalizes. When one eigenspace is too small, no rearrangement of eigenvectors fixes it. The missing dimensions live one step out, in the generalized eigenspace, and the chain length for that eigenvalue equals the size of its largest Jordan block. Gilbert Strang's MIT OpenCourseWare lecture on similar matrices and Jordan form develops exactly this picture: the Jordan form is the closest thing to diagonal that a defective matrix allows, and its blocks are read off from the chains.

Why chains matter: repeated roots in differential equations

Jordan chains earn their keep the moment a system of differential equations has a repeated eigenvalue. The extra vector in the chain is what produces the t·e^λt term in the solution.

For x′ = Ax with a simple eigenvalue, each eigenvector contributes a clean term c·e^λtv. With a defective double eigenvalue there is only one such term, yet the system still needs two independent solutions. The second one is built from the chain: (tv1 + v2)e^λt. Differentiate it and the chain relation (A − λI)v2 = v1 is exactly what makes it work. The same pattern drives matrix exponentials: on a Jordan block, e^At picks up polynomial factors in t, with the block size setting the highest power. The LibreTexts treatment of the Jordan canonical form shows the block structure behind this, and the LibreTexts appendix on diagonalization and linear systems walks from repeated eigenvalues to the Jordan form for differential equations specifically. Repeated roots show up in damped oscillators, control systems with coincident poles, and Markov chains with defective transition matrices, so this is a working tool, not a curiosity.

What competing calculators miss

For the exact phrase "generalized eigenvector calculator" there is no dedicated rival tool: the gap is that the computation exists online only under other names, or not at all.

• Symbolab and the WolframAlpha front end: strong eigenvalue and eigenvector solvers, but neither offers a generalized eigenvector or Jordan chain form. You can get v1 and then stop.
• Pearson's eigenvalue and eigenvector calculator: the strongest ordinary rival. It finds eigenvalues itself, solves for ordinary eigenvectors, and its examples even flag defective and diagonalizable cases using matrices like [[2, 1], [0, 2]]. It still produces no generalized eigenvector, so the flagged case is the one it cannot finish.
• dCode's Jordan Normal Form tool: the chain computation does happen here, buried inside a full Jordan decomposition. Its own worked example finds the extra vector by solving a chained system of the same shape as (A − λI)v2 = v1, but you asked for a chain and got a decomposition to unpack.
• The gap this page fills: a tool named for the job, which takes the matrix and λ, returns v1, v2 and v3 as vectors in their own right, states the chain length and geometric multiplicity, and prints the numeric check for every link so the answer can be verified on sight.

Limitations: what this calculator does not do

This tool builds one chain for one supplied eigenvalue of a real 2×2 or 3×3 matrix. It does not compute a full Jordan decomposition, and it does not find eigenvalues.

λ is an input, not an output, by design: the chain questions students and engineers bring here start from a known repeated eigenvalue. The entries are real numbers only. A matrix whose repeated eigenvalue is complex needs complex arithmetic throughout the chain; for a single complex eigenvector the complex eigenvector calculator covers that separate job. One subtler point: when the geometric multiplicity is 2 or more, several chains can exist side by side. This calculator tries a chain from every basis vector its row reduction produces and shows the longest one, so the reported chain length and the defective verdict reflect the size of the largest Jordan block; when several chains tie for longest, which one is shown can still depend on the starting basis vector. A full multi-chain analysis belongs to a Jordan decomposition tool such as the one described in the Wolfram MathWorld page on the Jordan canonical form. Last, the arithmetic is floating point with a 1e-10 pivot tolerance. Textbook matrices with integer or simple decimal entries, which are what chains are taught and tested on, come out exact to the 4 decimals shown; matrices with entries engineered to sit on the edge of singularity can make any floating point solver hesitate, and the verification rows are there so you can see it if it happens.

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