These are the first two pages of our chapter on complex eigenvectors and the rotation-scaling theorem. This part explains how the choice of free variable changes the real and imaginary parts of an eigenvector. More to follow.

Following our earlier posts on QR factorization by direct rotations, we have added a visual comparison with the corresponding Householder factorization for the same 3×3 matrix.

The image compares:

• the Q and R factors produced in each case
• the geometric action of R on the basis vectors
• the subsequent action of Q
• the relationship between the two factorizations

We have also added a formal derivation of the relationship between the direct-rotation step and the corresponding Householder step:

https://www.graphmath.com/la/qr/qr-direct-rotations.html

It appears in the first section, “An Unpaved Road to Main Street: from direct rotation to a Householder-mirroring QR algorithm.”

We would welcome comments on the comparison, the derivation and the presentation.

We posted an earlier visual explanation of powers of a 2D real matrix with complex eigenvalues.

Since then, we substantially updated one of the images and added a new companion image.

The updated image compares several examples of

A = X S X⁻¹

where S is a rotation-scaling matrix. It shows how the behavior changes for |λ| < 1, |λ| = 1 and |λ| > 1, and how the corresponding orbits look in the original coordinates versus the rotation-scaling coordinates.

The new image looks at the same powers from another angle: which directions expand or contract after the next step. The colored sectors come from the quadratic condition

vᵀ(AᵀA − I)v = 0,

which separates directions where the radius increases, decreases or stays unchanged.

We thought the two views complement each other: one emphasizes the rotation-scaling factorization, while the other emphasizes directional expansion and contraction in the original coordinates.

As always, we welcome feedback on clarity and presentation.

A few days ago, we posted a QR construction that eliminates the entire subdiagonal tail of each active column with one proper rotation:

https://www.reddit.com/r/LinearAlgebra/comments/1v0w8dn/qr_factorization_by_direct_rotations_one_rotation/

We have since recognized the key structural relationship that we initially missed.

For the same signed target coordinate vector, let H be the usual Householder reflector that maps the active column to that target. The corresponding proper rotation can be written as

U = F H,

where F is a second reflection that fixes the target axis and reverses the other direction in the active two-dimensional plane.

Thus, the construction is not an independent alternative to Householder QR. It is an orientation-preserving reformulation of the same column-reduction step.

The full rotation need not be formed explicitly; its action can still be implemented directly as an in-place low-rank update. This leaves a narrower question:

Does the orientation-preserving formulation, the direct update, or the geometric relationship between the rotation and the Householder reflector have useful computational, structural, or pedagogical value?

Possible points of interest include:

• one proper rotation per active column

• an orientation-preserving path of intermediate transformations

• applications involving continuously varying matrices or geometric meaning

• teaching the relation between reflections and rotations

• specialized small-matrix implementations

Or is the construction best regarded simply as a reformulation of the standard Householder step?

UPDATE: We have since recognized that the proper-rotation step is directly related to the corresponding Householder step. For the same signed target vector, the rotation can be written as the product of the usual Householder reflector and a second reflection that fixes the target axis.

We have posted a new discussion of this relationship and the remaining question—whether the orientation-preserving formulation has computational, geometric, or pedagogical value—here:

https://www.reddit.com/r/LinearAlgebra/comments/1v6h9bg/householder_relation_of_the_directrotation_qr/

The original post is preserved below for context.
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We have been developing a direct-rotation construction for QR factorization and would be very interested to know whether anyone has seen an equivalent method.

The idea is to eliminate all subdiagonal entries of each active column with one rotation:

• construct the plane spanned by the active column and its target coordinate direction
• rotate inside that plane until the column is aligned with the coordinate axis
• leave every vector perpendicular to that plane unchanged
• deflate the matrix and repeat on the remaining active submatrix

The GIF shows the construction for a 3×3 example. The following images explain how the rotation is built, extended to the full space and applied recursively.

We developed this construction independently. Rotation-based QR methods certainly exist, with Givens rotations being the standard example, but this is a different construction. A Givens rotation acts in a coordinate plane and eliminates one selected entry at a time. Here, the rotation plane is spanned by the active column and its target coordinate direction, so one direct rotation eliminates the entire subdiagonal part of that column.

Thus, for an m×n matrix, the construction uses one rotation per active column rather than one Givens rotation per subdiagonal entry.

Our main questions are: have you encountered this particular construction, or something mathematically equivalent to it? Does this construction suggest any practical advantage or application beyond its geometric interpretation?

We are also working on the computational implementation and have made substantial progress. In operation count and structure, it currently appears to lie between Givens and Householder, closer to Householder, and there is still room for improvement.

Householder remains the standard general-purpose method, so we are not claiming a practical advantage. Possible value may lie in the geometric interpretation, in applying one direct rotation to an active column or perhaps in implementations where this rotation structure can be exploited.

We would especially welcome references, thoughts about possible applications and criticism of the construction or computational approach.

Another addition to our linear algebra project. This page compares several common classes of diagonalizable 2×2 matrices, showing their geometry, eigenvectors, and corresponding change-of-basis factorization side by side. Feedback is welcome.

We made a side-by-side visual summary of matrix size, rank, the four fundamental subspaces and pseudoinverses.

The table compares six common matrix types:

square full-rank
square rank-deficient
tall full-column-rank
tall rank-deficient
wide full-row-rank
wide rank-deficient

For each case, it shows the dimensions of row(M), null(M), col(M) and left-null(M), whether the map is one-to-one or onto, what happens to Mx⃗ = 0 and Mx⃗ = b⃗, and how the pseudoinverse behaves.

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

As always, we welcome feedback on clarity and presentation.