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Eigen Vectors


Linear Algebra·EigenvectorsAn eigenvector is a direction A leaves alone — only stretches by some factor λ. Drag v around. When v and Av line up, you've found one. Eigenvalue is |Av|/|v| (signed).
λ=2λ=0.5Avv
v=(2, 1)
Av=(4, 0.5)
▸Animation
0.00 / 1.00

Diagonal matrix

2 distinct real · λ = 2, 0.5

A diagonal matrix scales each axis independently. The standard basis vectors are themselves eigenvectors: ê₁ stretches by 2, ê₂ shrinks by 0.5. The eigendirections are exactly the coordinate axes.
Learn more about distinct real eigenvalues · all four categories

01Livedrag v

|v|2.24|Av|4.03|Av| / |v|1.8angle(v, Av)-19.44°
Drag v until Av aligns with it.

02Eigen structure of A

λ1=2v(1, 0)
λ2=0.5v(0, 1)
λ² − 2.5·λ + 1 = 0  ·  Δ = 2.25

Key Terms
Getting Started
Dragging v and Watching Av
Preset Scenarios — Four Categories
Distinct Real Eigenvalues: Two Fixed Directions
Repeated Eigenvalue With a Full Set of Directions
Defective: a Repeated Eigenvalue Short of Directions
Complex Eigenvalues: No Real Direction Survives
The Alignment Signal and Eigenvalue Readout
The Eigen Structure Card
Using the Snap Button
Display Layer Toggles
What Is an Eigenvector?
Four Cases: Distinct, Isotropic, Defective, Complex
Related Concepts






Key Terms

Eigenvector — A nonzero vector vv such that Av=λvAv = \lambda v for some scalar λ\lambda. Matrix AA leaves the direction of vv unchanged and only scales it.

Eigenvalue — The scalar λ\lambda in Av=λvAv = \lambda v. It is the stretch factor along the eigenvector direction. A negative eigenvalue means the vector flips.

Eigendirection — The line through the origin spanned by an eigenvector. Every vector on this line is also an eigenvector with the same eigenvalue.

Characteristic Polynomial — The quadratic λ2−tr(A)λ+det⁡(A)=0\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0 whose roots are the eigenvalues of a 2×22 \times 2 matrix AA.

Defective Matrix — A matrix with a repeated eigenvalue but fewer eigenvectors than the algebraic multiplicity. Cannot be diagonalized.

Spectral Theorem — Every symmetric matrix has perpendicular eigenvectors and real eigenvalues.

Getting Started

The canvas shows the standard grid in gray, dashed green lines marking the eigendirections of AA, the test vector vv in orange, and its image AvAv in cyan. You control vv; the matrix AA comes from the scenarios panel on the left.

The mission is simple: drag vv around until it lines up with AvAv. When that happens, vv is an eigenvector and the ratio ∣Av∣/∣v∣|Av|/|v| is its eigenvalue λ\lambda.

Three quick experiments:

• Drag vv off the green dashed line — AvAv swings to a different direction. They do not match.
• Drag vv onto the green dashed line — the two vectors snap to amber and the status strip announces the eigenvalue.
• Drag vv to the perpendicular eigendirection (in the diagonal preset) — same alignment, different λ\lambda.

The green dashed lines are cheat sheets. The whole canvas is a search for the moments when vv and AvAv are collinear.

Dragging v and Watching Av

Only the orange vv handle is draggable. Everything else — AvAv, the angle arc, the alignment color — updates instantly.

• Drag vv in a circle — watch AvAv trace its own loop. The two curves agree only at the eigendirections.
• Speed of AvAv relative to vv — in stretchy directions AvAv runs ahead; in shrinking directions it lags behind. The eigenvalue magnitude is exactly that speed ratio.
• Drag vv near the origin — both vectors shrink. The ratio ∣Av∣/∣v∣|Av|/|v| stays defined as long as vv is nonzero.

A faster diagnostic than chasing alignment by eye: watch the angle readout. As you approach an eigendirection, angle(v,Av)\text{angle}(v, Av) heads to 0deg⁡0\deg or 180deg⁡180\deg. The status strip flips when the alignment is within roughly four degrees.

Preset Scenarios — Four Categories

The scenarios panel on the left jumps the matrix AA to eleven canonical examples grouped by eigen structure.

• Two distinct real — diagonal (λ=2,0.5)(\lambda = 2, 0.5), symmetric (λ=3,1)(\lambda = 3, 1) with perpendicular eigenvectors, reflection across y=xy=x (λ=1,−1)(\lambda = 1, -1), upper-triangular (λ=3,2)(\lambda = 3, 2) with non-orthogonal eigenvectors.
• Isotropic — identity (every direction, λ=1\lambda = 1) and uniform scaling by 2 (every direction, λ=2\lambda = 2).
• Defective — shear λ=1\lambda = 1 doubled, and defective λ=2\lambda = 2 doubled. Only one eigendirection in each.
• Complex — rotation by 30deg⁡30\deg, quarter turn (λ=±i)(\lambda = \pm i), and rotating spiral. No real eigendirection in any of them.

The explanation card at the top of the right column updates with a brief description of the structural feature for each preset.

Distinct Real Eigenvalues: Two Fixed Directions

Four of the eleven presets have two different real eigenvalues. The frozen picture below is the diagonal matrix egin{pmatrix} 2 & 0 \ 0 & 0.5 end{pmatrix}, whose eigenvalues are 22 and 0.50.5.

Two dashed eigenlines are drawn, one per eigenvalue. Any vector lying along one of them is mapped to a multiple of itself: the transformation stretches it or shrinks it but never turns it.
λ=2λ=0.5Avv
Distinct real eigenvalues, frozen

The diagonal matrix, eigenvalues 2 and 0.5. Two dashed eigenlines - the coordinate axes here - and a probe vector v whose image Av is turned away from it because v sits on neither line.

This is the case the whole tool is calibrated around, and it is the easiest to read. Drag vv onto an eigenline and the alignment signal fires — AvAv becomes parallel to vv, the angle arc collapses to zero, and the eigenvalue readout shows the factor.

Two independent eigendirections mean the matrix is diagonalisable: written in the basis of its own eigenvectors, it becomes diagonal, and applying it repeatedly just raises each eigenvalue to a power. That is why distinct real eigenvalues are the comfortable case in almost every application.

Note that "distinct" refers to the eigenvalues, not the directions. The reflection preset has eigenvalues 11 and −1-1 — still distinct, still two clean eigenlines, but one direction is flipped rather than stretched.

Repeated Eigenvalue With a Full Set of Directions

Two presets have a single repeated eigenvalue and still behave perfectly well: the identity, and the 2imes2 imes scaling shown below.

For A=2IA = 2I the only eigenvalue is 22, but every direction is an eigendirection. Drag vv anywhere at all and AvAv stays parallel to it — the alignment signal never switches off.
every direction · λ=2Avv
Repeated eigenvalue, full eigenspace, frozen

The 2x scaling. One eigenvalue, but every direction is an eigendirection, so v and Av are parallel wherever v is placed and no eigenline is singled out.

These are the isotropic maps: they scale the whole plane by one factor, treating every direction identically. The tool draws no distinguished eigenlines here because there is nothing to distinguish — the eigenspace is the entire plane rather than a line.

The important point is that a repeated eigenvalue is not by itself a problem. What matters is whether the eigenspace is big enough: here the eigenvalue 22 repeats twice and comes with a two-dimensional eigenspace, so the matrix is still diagonalisable. The next category is what happens when that fails.

Defective: a Repeated Eigenvalue Short of Directions

Two presets are defective. The frozen picture shows one: a repeated eigenvalue whose eigenspace is only one-dimensional, so a single eigenline appears where the previous category had a whole plane.

Drag vv anywhere off that one line and AvAv refuses to align, no matter where you put it.
λ=2 (only)Avv
Defective matrix, frozen

A repeated eigenvalue with only a one-dimensional eigenspace: a single eigenline where the previous case had the whole plane. Off that line, Av never aligns with v.

This is the case that breaks diagonalisation. An nimesnn imes n matrix needs nn independent eigenvectors to be written in a basis of its own eigenvectors, and a defective matrix simply does not have them — the algebraic multiplicity of the eigenvalue exceeds its geometric multiplicity.

The shear is the classic example. Every vector along the shear axis is fixed, but nothing else keeps its direction, so there is one eigenline where there ought to be two. Such matrices are handled with the Jordan form instead of a diagonal one, and they are the reason "every matrix is diagonalisable" is false.

Complex Eigenvalues: No Real Direction Survives

Three presets have complex eigenvalues, and the tool draws no eigenlines at all for them. The frozen picture is a 30° rotation.

Drag vv anywhere on the plane and AvAv never lines up with it. The angle arc between the two arrows never closes, because a rotation genuinely turns every direction.
Avv
Complex eigenvalues, frozen

A 30 degree rotation. No eigenlines are drawn at all, and the angle arc between v and Av never closes - no real direction is preserved.

That is the honest geometric content of a complex eigenvalue: no real line through the origin is preserved. The eigenvalues still exist — they come as a conjugate pair apmbia pm bi — but their eigenvectors live in mathbbC2mathbb{C}^2 and have no real representative to draw.

The pair still carries the geometry. Its modulus is the scaling factor per application, so ∣lambda∣=1|lambda| = 1 means a rigid rotation while ∣lambda∣>1|lambda| > 1 produces the outward spiral of the third preset; its argument is the rotation angle. Reading a rotation as "complex eigenvalues" rather than "no eigenvalues" is what keeps the theory uniform: over mathbbCmathbb{C}, every 2imes22 imes 2 matrix has two eigenvalues counted with multiplicity.

The Alignment Signal and Eigenvalue Readout

Card 01 (Live) shows four numbers updating in real time.

• ∣v∣|v| — length of your test vector.
• ∣Av∣|Av| — length of its image. Larger when AA stretches in this direction, smaller when it shrinks.
• ∣Av∣/∣v∣|Av|/|v| — the candidate eigenvalue magnitude. Only equals ∣λ∣|\lambda| exactly when vv is an eigenvector.
• angle(v,Av)(v, Av) — the smoking gun. At zero or 180 degrees, vv is parallel to AvAv, so you have an eigenvector.

When alignment is detected, the status strip turns amber and reads v is an eigenvector · λ ≈ (value). The sign of λ\lambda comes from dot(v,Av)\text{dot}(v, Av): positive when they point the same way, negative when opposite (reflection-type eigendirection).

The Eigen Structure Card

Card 02 (Eigen structure of A) shows all eigenvalues of the current matrix, plus a unit eigenvector beside each, and the characteristic polynomial at the bottom.

Four possible appearances:

• Two distinct real — rows λ1\lambda_1 and λ2\lambda_2 in green, each with its unit eigenvector (x,y)(x, y). Two green dashed lines on the canvas.
• Defective — one row in pink with the repeated λ\lambda and a single eigenvector, plus a note that a generalized eigenvector is required for a full basis.
• Isotropic — one row in green: λ\lambda with the message "every direction is eigen". On the canvas, concentric green rings replace the lines.
• Complex — one row in purple: λ=a±bi\lambda = a \pm bi with the note "no real eigenvector".

The bottom line shows λ2−tr(A)λ+det⁡(A)=0\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0 with the actual trace and determinant filled in, plus the discriminant Δ\Delta.

Using the Snap Button

The green button at the bottom of the eigen structure card jumps vv directly to the nearest eigendirection.

• Distinct case — snaps to whichever of the two eigenlines is closer to vv, preserving the current length of vv.
• Defective case — snaps to the single eigendirection that exists.
• Isotropic case — leaves vv unchanged (any direction already qualifies).
• Complex case — button is disabled and reads "No real eigenvectors" because no direction satisfies Av=λvAv = \lambda v in real two-dimensional space.

This is the fastest way to confirm what the eigen structure card claims: hit snap, watch vv lock to a dashed green line, see the alignment indicator turn amber, and read λ\lambda in the live card.

Display Layer Toggles

The chip strip toggles which visual elements appear on the canvas.

• grid — gray standard grid. Off for a cleaner background.
• eigenlines — the dashed green lines marking all real eigendirections. Turn off to hunt for them by alignment alone.
• v / Av lines — thin dashed extensions of vv and AvAv through the origin. Helps eyeball alignment when the vectors are short.
• angle arc — the small arc between vv and AvAv. Visual companion to the angle readout in the live card.
• labels — the vv, AvAv, and λ\lambda name tags. Off for screenshots or clean exploration.

Useful pairing: turn off eigenlines and try to find them yourself, then turn them back on to check your guesses.

What Is an Eigenvector?

Given a square matrix AA, an eigenvector is a nonzero vector vv that is mapped to a scalar multiple of itself:

Av=λvA v = \lambda v


The scalar λ\lambda is the eigenvalue. Geometrically, AA does many things to most vectors — rotates them, shears them, mixes their components — but along an eigenvector it does only one thing: stretch or shrink. The direction is invariant.

This is why eigenvectors matter: they reveal the natural axes of a linear transformation. In coordinates aligned with the eigenvectors, AA becomes a diagonal matrix, which is the simplest description possible.

For deeper coverage of definitions, properties, and applications, see eigenvectors theory page, eigenvalues definition, and matrix diagonalization.

Four Cases: Distinct, Isotropic, Defective, Complex

Every 2×22 \times 2 matrix falls into one of four structural categories.

• Two distinct real eigenvalues — the generic case. Two independent eigendirections. AA is diagonalizable. Symmetric matrices always land here and additionally have perpendicular eigenvectors.
• Isotropic (repeated, diagonalizable) — AA is a scalar multiple of the identity. Every direction is an eigendirection with the same λ\lambda.
• Defective (repeated, not diagonalizable) — the eigenvalue repeats but only one eigenvector exists. Classic example: a shear. Diagonalization fails; Jordan form is needed.
• Complex conjugate pair — eigenvalues are a±bia \pm bi with b≠0b \neq 0. AA has a rotational component. No real direction is preserved.

The discriminant Δ=tr(A)2−4det⁡(A)\Delta = \text{tr}(A)^2 - 4\det(A) in the characteristic polynomial diagnoses the case: positive means distinct real, zero means repeated, negative means complex.

For full treatment see characteristic polynomial, defective matrices, and complex eigenvalues.

Related Concepts

Eigenvalues — the scalar stretch factors that pair with each eigenvector.

Characteristic Polynomial — the source equation det⁡(A−λI)=0\det(A - \lambda I) = 0 whose roots are the eigenvalues.

Diagonalization — rewriting AA as PDP−1PDP^{-1} where DD is diagonal and PP has eigenvectors as columns.

Spectral Theorem — guarantees perpendicular eigenvectors and real eigenvalues for symmetric matrices.

Singular Value Decomposition — a generalization of eigendecomposition that works for any matrix, not just square diagonalizable ones.

Change of Basis — expressing AA in the eigenbasis turns it into a diagonal matrix.

Matrix Trace and Determinant — the two invariants that appear in the characteristic polynomial.