u/LinearAlgebraWorld

We made a visual explanation of powers of a real 2×2 matrix with complex eigenvalues.

For such a matrix, we can write

A = X S X⁻¹

where S is a rotation-scaling matrix. Then powers are computed as

Aᵗ = X Sᵗ X⁻¹.

The idea is that Sᵗ is easy to understand geometrically: it rotates by tθ and scales by |λ|ᵗ. The change of basis by X and X⁻¹ turns this circular rotation-scaling picture into the ellipse-like spirals seen in the original coordinates.

The first image follows one example through the factorization. The second shows more numerical examples with |λ| < 1, |λ| = 1 and |λ| > 1.

As always, we welcome feedback on clarity and presentation.

We previously posted separate 3D animations of QR decomposition by Gram–Schmidt, Givens rotations and Householder reflections.

Here is a 2D comparison of the same three geometric routes.

The goal is to show that all three methods reach the same kind of result, A = QR, but by very different geometric actions:

1. Gram–Schmidt: subtract projections and normalize
2. Givens rotations: rotate selected components to zero
3. Householder reflections: reflect vectors across chosen lines or planes

In 2D, the Givens case is almost trivial: only one rotation is needed to zero the lower-left entry. In higher dimensions, Givens QR proceeds by many such rotations, one entry at a time.

One detail worth noticing: Gram–Schmidt, in its standard form, produces positive diagonal entries in R. Givens and Householder versions may produce different signs depending on rotation/reflection sign choices. This is normal: QR is unique only after an extra sign convention is imposed, such as requiring positive diagonal entries in R.

More explanation and the 3D versions are here:
https://www.graphmath.com/la/visuals/qr/qr-three-geometric-routes.html

We will also add these 2D animations to that page.

We welcome feedback on clarity and presentation.

We made three animated comparisons showing how eigenvectors and eigenvalues behave across different families of 2D linear transformations.

1. Non-symmetric matrices with real eigenvalues, where the eigenvector directions need not be perpendicular
2. Symmetric matrices with real eigenvalues, where the eigenvector directions are orthogonal
3. Real matrices with complex eigenvalues, where no nonzero real direction remains on the same line

In each animation, the transformation develops continuously from the identity matrix to the displayed matrix. The goal is to make the difference between these three cases visible rather than only algebraic.

Full-size animations and explanations:
https://www.graphmath.com/la/visuals/eigenvectors-eigenvalues-2d-transformations.html

As always, we welcome feedback on clarity and presentation.

A visual definition of eigenvectors, followed by examples across several common 2D transformations.

The table compares uniform and non-uniform scaling, shear, triangular and symmetric matrices, projection, reflection, rotation and related cases. It shows which directions remain on the same line after transformation, their corresponding eigenvalues, and when no real eigenvectors exist.

Hope you don’t mind the size and density — we wanted to keep all the examples together so they could be compared directly. Opening the image at full size is recommended.

The goal is to make the definition A x⃗ = λx⃗ visible across many different matrix types.

We welcome feedback on clarity and presentation.

UPDATE: we have added a few more examples, you can see the full updated version on our web-site:
https://www.graphmath.com/la/visuals/eigenvectors-definition-and-examples.html

We made a visual derivation of least squares from the original overdetermined system.

The first image shows how projecting the data vector b onto the column space of A leads to the normal equations

AᵀAβ = Aᵀb

and gives the best-fit line y = β₁ + β₂x.

The second keeps the same data but changes the model to y = β₁ + β₂x². The second column of A changes from x to x², so the prediction plane changes, but the same projection method applies.

The point is to show least squares as geometry rather than a formula to memorize.

As always, we welcome feedback on clarity and presentation.

We made a paired visual derivation of orthogonal projection, starting from its defining condition:

the projection lies in the target subspace, and the leftover residual is orthogonal to that subspace.

For projection onto a single vector, this immediately gives the familiar scalar projection formula. Replacing the single vector by the columns of a matrix gives the same derivation for

P = U(UᵀU)⁻¹Uᵀ.

The point is not just to know the formulas, but to be able to reconstruct them instead of memorizing them.

As always, we welcome feedback on clarity and presentation.

Made this 2D version after apt and kind suggestion of u/strange-the-quark and watching relevant 3Blue1Brown video.

We made a visual derivation of Cramer’s rule using signed volumes.

The image shows the 3D case: the denominator is the signed volume of the parallelepiped spanned by the columns of A, while the numerator replaces one column by b. By projecting both shapes onto the same normal direction, the ratio of volumes becomes the corresponding coordinate xᵢ​.

A related page with the full visual explanation, including both the 3D derivation and a compact ℝⁿ version, is here:
https://www.graphmath.com/la/visuals/cramers-rule-geometric-derivation.html

We welcome feedback on clarity and presentation.

The determinant permutation list can be built recursively:

choose the first-row entry, then permute what remains.

The “what remains” part is exactly the permutation list of the corresponding submatrix.

So the same recursive object is called Sₙ​ in the full determinant, and Sₙ₋₁ inside each cofactor submatrix.

4×4 version:
www.graphmath.com/la/figures/determinant/permutations4x4.png

We made an animated visualization of QR decomposition using Householder reflections.

This uses the same matrix as our earlier Gram–Schmidt and Givens rotation QR animations, so the three methods can be compared on the same example.

Householder QR reaches an upper triangular matrix by reflecting the active part of the matrix column by column. In this 3×3 example it takes two reflections, compared with three Givens rotations, which eliminate one entry at a time.

A related page with another Householder QR example is here:
https://www.graphmath.com/la/visuals/householder-reflection.html

We welcome feedback on clarity, pacing and presentation.

We made an animated visualization of QR decomposition using Givens rotations.

This uses the same matrix as our earlier Gram–Schmidt QR animation, so the two methods can be compared directly.

The animation shows how successive rotations zero out entries below the diagonal while preserving lengths and angles. In 3D, each Givens rotation is a rotation in one coordinate plane, so the columns visibly move until the matrix becomes upper triangular:

A = QR

This is the same QR goal as Gram–Schmidt, but the geometric action is different: instead of projecting and subtracting components, Givens rotations turn vectors in coordinate planes until selected components disappear.

For this animation and a related Givens rotation visualization, see:
https://www.graphmath.com/la/visuals/givens-rotation-algorithm.html

We welcome feedback on clarity, pacing and presentation.

We made an interactive visualization of Gram–Schmidt as QR decomposition:
Gram Schmidt Stepwise Visualization

It shows each column being projected, corrected, normalized, and added to Q, while the corresponding coefficients form R. The goal is to make the geometry behind the usual algebraic steps visible: projections, orthogonal components, scaling, and the final A = QR.

We welcome feedback on presentation and clarity

I made a visualization of cofactor expansion for a 3×3 determinant.

The idea is to start with the usual interpretation of det(A) as signed volume, then show each cofactor term as:

• one coordinate projection of the expansion column

• one projected 2×2 minor in the orthogonal coordinate plane

So instead of treating

det(A) = a₁₁M₁₁ − a₂₁M₂₁ + a₃₁M₃₁

as only a symbolic rule, the goal is to make the three terms look like signed volume pieces.

The image shows the three terms for one example matrix, including the “same sign / negated sign” behavior of the cofactor signs.

Full explanation, including determinant sign:

https://www.graphmath.com/la/determinant/determinant-sign.html

I would be interested in feedback on whether the geometric interpretation is clear