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Jordan Normal Form Calculator (J, P and block structure)

Exact Jordan form of a 1×1 to 6×6 matrix: the blocks from the ranks of (A−λI)ᵏ, the transition matrix P built from generalized eigenvector chains, and A·P = P·J checked exactly. When the characteristic polynomial does not split over ℚ it says so instead of guessing.

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Jordan Normal Form Calculator (J, P and block structure) is free to use as often as you like, directly from this page. Its place is under Matrices & vectors; RREF Calculator (Reduced Row Echelon Form) and Normal Distribution Calculator answer the questions closest to this one.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Jordan Normal Form Calculator (J, P and block structure)?

Exact Jordan form of a 1×1 to 6×6 matrix: the blocks from the ranks of (A−λI)ᵏ, the transition matrix P built from generalized eigenvector chains, and A·P = P·J checked exactly. When the characteristic polynomial does not split over ℚ it says so instead of guessing.

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is Jordan Normal Form Calculator (J, P and block structure) different from RREF Calculator (Reduced Row Echelon Form)?

They sit next to each other but answer different questions. RREF Calculator (Reduced Row Echelon Form): gauss-Jordan elimination up to 10×10 in exact rational arithmetic, with every row operation written out as R₂ ← R₂ − 2R₁ and the matrix redrawn after each one. Augmented mode classifies the system and gives the solution set. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Normal Distribution Calculator is the closest one after this: Compute normal probabilities P(X≤x), P(X>x), P(a≤X≤b) or find x for a percentile, from the mean and standard deviation.

What else is worth having open alongside it?

Point-Slope Form Calculator and Slope-Intercept Form Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

All guides →
ExplainerThe Expected Value Is a Number That Never HappensOutcomes of 10, 20 and −5 with probabilities 0.5, 0.3 and 0.2 give an expected value of 10. Shift the probabilities slightly and it becomes 10.5 — a result none of the three outcomes can produce.ExplainerAn Error Bar That Shrinks Is Not the Standard DeviationFor 10, 12, 15, 11, 13, 14 the standard deviation is 1.87 and the standard error is 0.76. They answer different questions, and only one of them gets smaller as you collect more data.ExplainerAn r of 0.85 That Proves NothingFive points give r = 0.853, which sounds strong. The threshold for significance at five points is 0.878, so this one falls short — and a perfect parabola scores exactly zero.ExplainerQuadruple the Sample to Halve the IntervalA mean of 100 with a standard deviation of 15 over 30 observations gives 100 ± 5.37 at 95 %. Going to 120 observations gives ± 2.68 — exactly half, because the sample size enters under a square root.ExplainerThe Same Three Temperatures Give Three Different Coefficients of VariationFor 2, 4, 4, 4, 5, 5, 7, 9 the coefficient of variation is 42.76 %. It is the right tool for comparing spread across different scales — and completely wrong for any scale whose zero was chosen by convention.ExplainerThree Dice Cluster, and One Die Does Not3d6 and 1d20 have almost the same average and nothing else in common. Against a target of 12 the three dice are better, against 18 they are five times worse — and the calculator gives the exact figures rather than a simulation.