Eigenvalues and eigenvectors

What is this tool for?

An eigenvector of a square matrix is a direction that the matrix does not turn: it only stretches it, shrinks it or flips it. The eigenvalue says by how much. If A·v = 3·v, the vector v keeps its direction and every length along it is multiplied by 3. Most vectors change direction when a matrix multiplies them. Eigenvectors are the exceptions, which is why they show what the matrix really does.

These numbers are used in many fields. Engineers use them to find the natural frequencies of a vibrating structure and to tell whether a system is stable. Google-style ranking of web pages relies on an eigenvector of a matrix of links. Principal component analysis uses them to find the main directions in a set of data. In quantum mechanics, the possible results of a measurement are eigenvalues.

This calculator finds the eigenvalues and eigenvectors of a real matrix up to 8 × 8, then goes further with diagonalization, the Jordan form, the matrix exponential and the singular value decomposition. If you are a student, you can follow each step from the characteristic polynomial to the vectors and compare it with your own working. If you are a researcher, you get exact values when possible, exact multiplicities, and the SVD and exponential for testing a small case or your own code. The calculation runs in your browser. Results are indicative: for published work, confirm them with a second method.

How to use it

1. Set the size of the matrix and fill in the cells. Eigenvalues need a square matrix; only the SVD accepts a rectangular one.
2. Choose what you are looking for: eigenvalues and eigenvectors, diagonalization, Jordan form, the exponential or the singular value decomposition.
3. Read the result on the receipt. Values are exact whenever possible, such as (1 + √5)/2, and the ≈ sign marks a rounded numeric value.

How it is calculated

Everything starts from the equation A·v = λ·v, which can also be written (A − λI)·v = 0, where I is the identity matrix. A non-zero vector v can satisfy it only if A − λI is not invertible, meaning that its determinant is 0. This gives the characteristic polynomial p(λ) = det(λI − A). For an n × n matrix it has degree n, and its roots are exactly the eigenvalues.

The calculator gets this polynomial with exact fractions and factors it, which gives the exact multiplicity of each root. For each eigenvalue, it then solves (A − λI)·v = 0 by elimination, as for a linear system. The solutions are the eigenvectors, and the number of independent solutions is the dimension shown on the page.

An eigenvalue is exact when it is a fraction, such as 3 or 1/2, or when it comes from a factor of degree 2, such as (1 + √5)/2 or i. If a factor of degree 3 or more remains once the fractions are removed, its roots are calculated numerically and marked ≈. Eigenvectors are exact only for eigenvalues that are fractions: for (1 + √5)/2, the value is exact but the vector is numeric.

The exponential is always numeric, and so are the SVD vectors. Singular values stay exact when possible, such as √5.

Worked examples

Each calculation explained

Eigenvalues and eigenvectors

Run this first to understand a matrix. It gives the characteristic polynomial, each eigenvalue with its multiplicity and dimension, the eigenvectors, and whether the matrix is diagonalizable.

The polynomial is calculated with exact fractions, then factored. Fractions and factors of degree 2 give exact roots, and the others are numeric. Each eigenvalue then leads to the system (A − λI)·v = 0.

[(2, 0, 0) ; (0, 3, 4) ; (0, 4, 9)] gives λ³ − 14λ² + 35λ − 22 = (λ − 11)(λ − 2)(λ − 1), with the vectors (0, 1, 2), (1, 0, 0) and (0, −2, 1). [(1, 1) ; (1, 0)] gives λ² − λ − 1 and the exact values (1 ± √5)/2. [(0, −1) ; (1, 0)] gives λ² + 1 and the complex values ± i.

Diagonalization

This calculation diagonalizes A, writing A = P·D·P⁻¹. A diagonal matrix makes powers simple, since Aᵏ = P·Dᵏ·P⁻¹, and it separates the equations of a linear differential system.

The calculator checks exactly whether each eigenvalue has as many independent vectors as its multiplicity. If so, the eigenvectors become the columns of P and the eigenvalues the diagonal of D, in the same order. It then calculates P⁻¹.

For [(2, 0, 0) ; (0, 3, 4) ; (0, 4, 9)], D has 11, 2 and 1 on its diagonal, and the columns of P are (0, 1, 2), (1, 0, 0) and (0, −2, 1). The rotation [(0, −1) ; (1, 0)] is diagonalizable only over the complex numbers.

Jordan decomposition

Use this when a matrix is not diagonalizable. It writes A = P·J·P⁻¹, where J has non-zero values only on its diagonal and just above it. If the matrix is diagonalizable, J is simply D.

For each eigenvalue, the calculator measures the dimension of the kernel of (A − λI), then of (A − λI)², and so on until it reaches the multiplicity. The way these dimensions grow fixes the number and size of the blocks. Chains of vectors then form the columns of P.

[(3, 1, 0) ; (0, 3, 0) ; (0, 0, 5)] gives p(λ) = (λ − 5)(λ − 3)². For λ = 3, the dimensions are 1 then 2, so there is one block of size 2 × 2. With the 1 × 1 block of λ = 5, this gives J = [(5, 0, 0) ; (0, 3, 1) ; (0, 0, 3)].

Matrix exponential

It is mainly used for linear differential systems: the solution of x′ = A·x is x(t) = e^(tA)·x(0). For a given value of t, multiply A by t first, then calculate the exponential.

The definition is the series e^A = I + A + A²/2! + A³/3! + …. In practice, the calculator divides A by 2 to the power s, applies a Padé approximant, then squares the result s times, with s chosen from ‖A‖₁. The result is always numeric and rounded for display.

For [(0, 1) ; (−1, 0)], e^A = [(cos 1, sin 1) ; (−sin 1, cos 1)], a rotation of 1 radian, with cos 1 ≈ 0.540302 and sin 1 ≈ 0.841471. You can check these two numbers with the scientific calculator, in radians.

Singular value decomposition

It works on any matrix, even a rectangular one, and writes A = U·Σ·Vᵀ. It gives the rank and the main directions of a data table. It is the usual tool behind principal component analysis, least squares and data compression.

The calculator forms the symmetric matrix A·Aᵀ, or Aᵀ·A if A has more rows than columns. The singular values are the square roots of its eigenvalues, largest first. Jacobi rotations then make the columns orthogonal and give U and V.

For [(3, 0) ; (0, 4)], A·Aᵀ = [(9, 0) ; (0, 16)] and p(λ) = λ² − 25λ + 144, so the singular values are 4 and 3. The 4 × 5 matrix [(1, 0, 0, 0, 2) ; (0, 0, 3, 0, 0) ; (0, 0, 0, 0, 0) ; (0, 2, 0, 0, 0)] has singular values 3, √5, 2 and 0, so its rank is 3.

Names and notation

• Eigen: a German word meaning “own”, in the sense of “belonging to”. An eigenvector in a way belongs to the matrix itself. English kept the German word, French translates it as “propre” and Arabic says “ذاتية”, “of itself”. David Hilbert used this German term in the early 20th century.
• λ: the Greek letter lambda, the usual symbol for an eigenvalue. The index in λ₁ or λ₂ only numbers them.
• v: the eigenvector, from the first letter of “vector”. The definition requires v ≠ 0.
• λI: the identity matrix I multiplied by λ, that is, λ on the diagonal and 0 elsewhere.
• Characteristic polynomial, p(λ): “characteristic” because it carries the eigenvalues of the matrix and does not change when the basis changes.
• Algebraic multiplicity: the number of times an eigenvalue is a root of p(λ). The page simply says “multiplicity” and writes m in the steps.
• Geometric multiplicity: the number of independent eigenvectors for an eigenvalue. The page calls it “dimension”. It is at least 1 and at most the algebraic multiplicity.
• P and D: D is the diagonal matrix of eigenvalues, hence its letter. The columns of P are the eigenvectors: it is the change-of-basis matrix.
• Jordan form, J: named after Camille Jordan, a French mathematician of the 19th century. A Jordan block has one eigenvalue on its diagonal and 1s just above it. Do not confuse him with Wilhelm Jordan, a German geodesist, who is the Jordan of Gauss–Jordan elimination.
• ker: short for the English word “kernel”, the set of vectors that a matrix sends to 0. The steps show dim ker(A − λI)², the dimension of the kernel of (A − λI)².
• e^A: the exponential of the matrix A, read “e to the A”. The number e is Euler's number, about 2.718282, and the series of the ordinary exponential extends to matrices this way.
• Padé approximant: named after Henri Padé, a French mathematician who studied these approximations of a function by a fraction of two polynomials.
• Singular value, σ: the Greek letter sigma, not to be confused with the σ of standard deviation. σ₁ is the largest: it is the strongest stretch the matrix applies to a vector of length 1.
• U, Σ and Vᵀ: Σ, capital sigma, has the size of A and holds the σ values on its diagonal. U and V are orthogonal matrices, and their columns are the singular vectors. The T means “transpose”: the rows of Vᵀ are the columns of V.
• SVD: the initials of the English “singular value decomposition”, also used in French and Arabic.
• Jacobi rotations: named after Carl Gustav Jacob Jacobi, a German mathematician of the 19th century, who used rotations in a plane to calculate the eigenvalues of symmetric matrices.

Tips and limitations

• Quick check: the sum of the eigenvalues, counted with their multiplicity, equals the trace (the sum of the diagonal), and their product equals the determinant. For [(2, 1) ; (1, 2)]: 3 + 1 = 2 + 2 and 3 × 1 = 3.
• To check an eigenvector, calculate A × v in the matrix calculator, then compare the result with λ·v.
• Type fractions rather than rounded decimals: 1/3, not 0.333. A repeated eigenvalue is fragile: replace the last 1 of [(1, 1) ; (0, 1)] with 1.001 and you get two different values, 1 and 1.001, and a matrix that is now diagonalizable.
• Limits: real cells only, up to 8 × 8 and 24 characters per cell. Complex eigenvalues are shown in the results, but you cannot type complex numbers or variables.
• The Display field switches between Exact values, Decimals and Significant digits, up to 15 digits. Only the display changes, not the calculation.
• For principal component analysis, first subtract from each column its mean, then run the SVD: the calculator does not centre the data for you, and it is limited to small tables, up to 8 × 8.

Mistakes to avoid

• Confusing the two multiplicities. In [(1, 1) ; (0, 1)], λ = 1 has multiplicity 2 but dimension 1: only the dimension counts independent vectors.
• Expecting one exact vector when the dimension is above 1. For the 3 × 3 identity matrix, λ = 1 has dimension 3: every non-zero vector works, and the calculator shows just one basis among many.
• Comparing SVD vectors sign by sign with another program. A column of U and the matching column of V can change sign together without changing A. Compare the singular values instead, or check that U·Σ·Vᵀ gives back A.
• Taking singular values for eigenvalues. The rotation [(0, −1) ; (1, 0)] has the eigenvalues i and −i, while its two singular values are both 1.
• Assuming there is an error because the polynomial has the opposite sign to your book. Some courses use det(A − λI): when n is odd, it is the opposite of det(λI − A), with the same roots.

Frequently asked questions

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