Linear Transformation and Eigenvector Visualizer

What is a 2D linear transformation matrix and determinant?

A 2D linear transformation maps each vector x to Ax through a 2×2 matrix A = [[a, b], [c, d]]. The determinant det(A) = ad − bc is the factor by which the transformation scales area, and its sign shows whether orientation is kept or flipped. Eigenvectors satisfy Av = λv: they stay on their own line through the origin and are only scaled by λ.

What a matrix does to the plane

A 2 × 2 matrix A = [[a, b], [c, d]] is a rule for moving every point of the plane: it sends (x, y) to (ax + by, cx + dy). Put in î = (1, 0) and out comes (a, c), the first column; put in ĵ = (0, 1) and out comes (b, d), the second. Every other point is x î + y ĵ, so it lands on the same mix of the two new arrows, x(a, c) + y(b, d). That is why grid lines stay straight, parallel and evenly spaced, why the origin never moves, and why the two columns, the two arrows you drag, are the whole matrix.

Using the visualiser

Drag the tip of î′ or ĵ′ and the matrix follows: the glyph in the corner prints the first column in î′’s colour and the second in ĵ′’s. Tips snap to multiples of 0.5; hold Shift, or turn snapping off, to place them anywhere, or type the four entries. Each preset under the stage is drawn as the shape it makes of the unit square, so a shear, a reflection or a projection is one tap away. Drag the probe v, or press anywhere on the plane to put it there, and Av is drawn beside it; when v lies on an eigenline, Av is a multiple of v and both light up. Play morphs the identity into A again. On the focused stage, 1, 2 and 3 pick î′, ĵ′ or v and the arrow keys move it by 0.1. The 3D tab takes a 3 × 3 matrix, and Then apply B composes two.

Worked example: [[1, 1], [0.5, 1.5]]

The matrix the visualiser opens on, worked by hand.

1. The columns. î lands on (1, 0.5) and ĵ on (1, 1.5), the two arrows.
2. The determinant. det A = ad − bc = 1 × 1.5 − 1 × 0.5 = 1. The unit square becomes a parallelogram of area 1, the same area, and the sign is positive, so orientation is kept.
3. The eigenvalues. The trace is a + d = 2.5, so the characteristic equation is λ² − 2.5λ + 1 = 0, which factors as (λ − 2)(λ − 0.5) = 0: λ = 2 and λ = 0.5.
4. The eigenvectors. For λ = 2, A − 2I = [[−1, 1], [0.5, −0.5]], whose rows are both perpendicular to (1, 1); for λ = 0.5, A − 0.5I = [[0.5, 1], [0.5, 1]], perpendicular to (2, −1). Check: A(1, 1) = (2, 2) = 2(1, 1), and A(2, −1) = (1, −0.5) = 0.5(2, −1). The two eigenlines meet at 71.57°, not 90°, because A is not symmetric.
5. The singular values. With s = a² + b² + c² + d² = 4.5, σ = √((s ± √(s² − 4 det²))/2) gives 2.065 and 0.4842, the semi-axes of the ellipse the unit circle becomes. Their product is 1, which is |det A|, as it must be.

To see the characteristic equation as a curve, plot y = x² − 2.5x + 1 in the graphing calculator: it crosses zero at 0.5 and 2, the two eigenvalues.

The determinant is an area

A sends the unit square, with corners 0, î, î + ĵ and ĵ, to the parallelogram with corners 0, (a, c), (a + b, c + d) and (b, d). Its area is |ad − bc|, and every other shape’s area is multiplied by the same factor, since any shape is made of small squares that all change alike. Scale by 2 makes det 4, twice as wide and twice as tall, and the shear [[1, 1], [0, 1]] has det 1: it slides the square into a parallelogram with the same base and height.

The sign carries orientation. When det is positive, ĵ′ is still anticlockwise of î′, which the arc at the origin shows; when it is negative, the plane has been turned over, as a mirror turns it, and the arc turns the other way, as the reflection in y = x does with det −1. When det is 0 the two columns lie on one line and the whole plane is flattened onto it, as projection onto the x-axis does, with det 0 and rank 1. The zero matrix, rank 0, goes further and sends the whole plane to the origin. Nothing can undo either.

That is also why Play sometimes passes through a flat grid. The morph follows the straight path M(t) = (1 − t)I + tA, and det M(t) = (1 − t + tλ₁)(1 − t + tλ₂), which reaches 0 on the way, at t = 1/(1 − λ), whenever an eigenvalue λ is negative. The reflection in y = x, with λ = −1, is flat at t = 0.5, and a rotation by 180° passes through the zero matrix there. A rotation by less than 180° has complex eigenvalues and never flattens.

Eigenvectors are the lines that stay put

An eigenvector is a nonzero v with Av = λv: A only stretches it, by the eigenvalue λ, so v stays on its own line through the origin. Rewritten as (A − λI)v = 0, it has a nonzero solution only when det(A − λI) = 0, which for a 2 × 2 matrix is the characteristic equation λ² − (a + d)λ + (ad − bc) = 0.

Its discriminant, tr² − 4 det, decides what you see. Positive, as for the default with 2.25, gives two real eigenvalues and two eigenlines. Zero gives one eigenvalue twice: either every direction is an eigenvector, as for 2I, or there is just one eigenline, as for the shear [[1, 1], [0, 1]], whose only eigenline is the x-axis. Negative gives a complex pair and no real eigenline at all. A rotation by 90° has λ² + 1 = 0, so λ = i and −i. [[1, −1], [1, 1]] has λ = 1 ± i, of modulus √2 = 1.414 and angle 45°, and it does exactly that: it turns every vector anticlockwise by 45° and scales it by 1.414.

Composition and order

Apply A and then B, and the single matrix that does both is BA, because B(Av) = (BA)v: the matrix next to the vector acts first. Order matters. A shear S = [[1, 1], [0, 1]] followed by a rotation R by 90° is RS = [[0, −1], [1, 1]], while the rotation followed by the shear is SR = [[1, −1], [1, 0]]. They are different maps, though they share their trace, 1, their determinant, 1, and their eigenvalues, 0.5 ± 0.866i, as BA and AB always do. Determinants multiply in either order, det(BA) = det B det A. The rotation matrix itself, [[cos θ, −sin θ], [sin θ, cos θ]], has as its columns the points at θ and θ + 90° on the unit circle explorer’s circle.

In three dimensions

A 3 × 3 matrix’s three columns are where î, ĵ and k̂ land, and the unit cube becomes the parallelepiped they span. Its signed volume is the determinant, the triple product Aî · (Aĵ × Ak̂), which is how the 3D vector and cross product explorer finds a box’s volume. The same product gives any crystal’s unit cell volume from its three edge vectors; for the cubic cells of the crystal lattice explorer it is simply a³.

The 3D tab opens on [[1, 0.5, 0], [0, 1, 0.5], [0.5, 0, 1]]. Its characteristic cubic is λ³ − 3λ² + 3λ − 1.125 = 0, which is (λ − 1)³ = 0.125, so λ = 1.5 along (1, 1, 1) and λ = 0.75 ± 0.4330i, of modulus 0.866 and angle 30°: it stretches the diagonal by 1.5 and turns the plane round it by 30° while shrinking it to 0.866. The determinant is 1.5 × 0.75 = 1.125. A cubic always crosses zero, so a real 3 × 3 matrix always has a real eigenvalue, which is why a rotation in space always has an axis: the rotation by 45° about z has λ = 1 along z and 0.7071 ± 0.7071i.

Common mistakes

• Reading the images off the rows. î lands on the first column, (a, c), not the first row. For [[1, 1], [0.5, 1.5]], î lands on (1, 0.5), not on (1, 1).
• Assuming eigenvectors are perpendicular. Only a symmetric matrix promises that. The default’s eigenlines, along (1, 1) and (2, −1), meet at 71.57°.
• Losing the sign of the determinant. Area scales by |det|, but the sign says whether the plane is turned over: the reflection in y = x has det −1, not 1.
• Writing AB for “A then B”. A then B is BA. The shear then the rotation is [[0, −1], [1, 1]]; written the other way round, [[1, −1], [1, 0]] is the rotation then the shear.