A linear map redrawing the plane

• the closed walks of length 1 are the 1th power sum of the eigenvalues ×8
• there are 2 graphs on 2 points up to relabelling ×5
• and the 5th power agrees with 5 multiplications ×2
• every determinant is a multiple of 2^(n−1) = 4 ×2
• the matrix is square, 2×2 to 7×7 ×2
• a centre neither collapses nor escapes ×1
• a component made of k discs holds exactly k eigenvalues ×1
• a degenerate sink carries a generic point to the origin ×1
• a graph with a cycle has at least as many edges as vertices ×1
• a map that keeps both dimensions has a non-zero determinant ×1
• a map that loses a dimension has determinant zero ×1
• a point outside every disc is not an eigenvalue ×1
• a power of the matrix is between 2 and 24 ×1
• a quoted maximum is a multiple of the unit and under the cap ×1
• a real 2×2 map has two eigenvalues counted with sign ×1
• a saddle carries a generic point away ×1
• a sink (node) carries a generic point to the origin ×1
• a sixth of the closed walks of length three is the number of triangles ×1
• a skew matrix exponentiates to a rotation, first entry ×1
• a source (node) carries a generic point away ×1
• a spiral sink carries a generic point to the origin ×1
• a spiral source carries a generic point away ×1
• a symmetric matrix's eigen-directions are perpendicular ×1
• a touching pair and a free pair were both found ×1
• adding a multiple of one column to another changes nothing ×1
• an ellipse of area past four times the determinant does hold a non-zero point ×1
• an odd number of grid lines between 3 and 9 ×1
• an odd number of grid lines between 3 and 9, so one runs through the point ×1
• an unsymmetric one's are not, which is what the panel is for ×1
• and agrees with the expansion along a row ×1
• and at a root the shifted map crushes something to nothing ×1
• and does not already point along the answer ×1
• and each direction really is left alone ×1
• and every pair running to the other finishes does cross ×1
• and every point at all lands on one line ×1
• and exp L has determinant one ×1
• and exponentiating the two eigenvalues gives the same matrix ×1
• and in some column disc ×1
• and it is at most the smaller of the two counts ×1
• and it keeps area exactly ×1
• and its discs obey the same count ×1
• and its signed area is about the determinant times the square's ×1
• and its size is the size of that number ×1
• and its two eigen-directions are genuinely different ×1
• and moves the permanent ×1
• and multiply to the determinant ×1
• and none passes the next one's starting value ×1
• and reaches it on the smallest patch drawn ×1
• and so are every two columns ×1
• and so are their images ×1
• and so does the triple product of the columns ×1
• and some non-embeddable matrices break the inequality ×1
• and the angle shrinks by the ratio of the two eigenvalues ×1
• and the boundary's image winds that many times round the origin ×1
• and the direction it occurs in is an eigen-direction, to within one sample ×1
• and the identity leaves the unit square alone ×1
• and the largest is the largest multiple of the unit under the bound ×1
• and the lowest at the second ×1
• and the lowest is the smaller ×1
• and the shortest is 1/√λ for the larger ×1
• and the shortest is the second ×1
• and the two add to the number of edges ×1
• and the two are a right angle apart ×1
• and there is a direction it sends to the origin ×1
• and where it has got to is what that rate predicts ×1
• any plane containing the top eigenvector reaches the top eigenvalue ×1
• AᵀA is symmetric and has two real eigenvalues ×1
• Av is λv in the first coordinate ×1
• Av is λv in the second ×1
• both eigenvalues are positive, so the level set is a closed curve ×1
• critical damping never overshoots ×1
• D⁻¹AD has the same eigenvalues as A ×1
• each coefficient of the characteristic polynomial is a whole number ×1
• each eigenvalue makes A − λI singular, by a determinant the polynomial never saw ×1
• each eigenvalue of the smaller block sits between two of the larger's ×1
• each exponentiates back to the same transition matrix ×1
• each Jacobi pair satisfies Av = λv ×1
• each marked matrix has determinant one ×1
• each matrix in a row is four finite entries under 8 in size ×1
• each root really is a root ×1
• every choice of struck row gives the same number ×1
• every corner of the image stays inside its panel ×1
• every eigenvalue lies in some row disc ×1
• every eigenvalue of a square and a lone point is a root of its characteristic polynomial ×1
• every eigenvalue of a star with four arms is a root of its characteristic polynomial ×1
• every eigenvalue of another graph on 6 points and 7 edges is a root of its characteristic polynomial ×1
• every eigenvalue of one graph on 6 points and 7 edges is a root of its characteristic polynomial ×1
• every embeddable matrix has 0 < det P ≤ p₁₁p₂₂p₃₃ ×1
• every entry is a sign ×1
• every matrix drawn has a real logarithm ×1
• every matrix lands inside the plane drawn ×1
• every one of these is a logarithm of the same matrix ×1
• every pair was classified once ×1
• every plane's maximum lies between the middle and top eigenvalues ×1
• every point drawn satisfies xᵀAx = 1 ×1
• every point of that line lands on the origin ×1
• every preimage maps back onto its target ×1
• every sign matrix is counted once ×1
• every start reaches every finish ×1
• every step brings the arrow nearer the dominant direction ×1
• every tree found has one edge fewer than it has vertices ×1
• every triangle has three different corners ×1
• every two rows are perpendicular ×1
• fourth ×1
• half the closed walks of length two is the number of edges ×1
• heavier damping settles more slowly ×1
• in both coordinates ×1
• inside the unit circle the series converges ×1
• near doing nothing, more matrices are embeddable than among matrices drawn at random ×1
• no angle's sine exceeds one ×1
• no determinant exceeds Hadamard's bound ×1
• no eigenvalue goes down ×1
• no exponential of a traceless matrix has trace below −2 ×1
• no triangle is listed twice ×1
• on a closed surface every edge is shared by exactly two triangles ×1
• on five points exactly two graphs share an adjacency spectrum: the star and the square with a lone point ×1
• one dimension survives and one is lost, and they add to the two started with ×1
• one eigenvalue is strictly the largest ×1
• one entry of v per row ×1
• one square on each side of the fold ×1
• one to three graphs ×1
• one to three spaces ×1
• one variant at a time ×1
• outside it the partial sums run away ×1
• P D P⁻¹ is the map it started from ×1
• Paley's matrix has size q + 1 for a prime q ≡ 3 mod 4 ×1
• reached at the top eigenvector itself ×1
• scaling one column scales the area by the same factor ×1
• second ×1
• size five stops at forty-eight ×1
• size four reaches the bound exactly ×1
• size three stops at four ×1
• so its nullity is the number of independent cycles, counted from the graph itself ×1
• so relative to the patch it falls by two ×1
• so the determinant counts the pairs that never meet ×1
• swapping two columns turns the sign over ×1
• Sylvester's matrices have sizes 2, 4, 8, 16 ×1
• t runs up to at most 40 ×1
• the alternating sum of the holes is the alternating sum of the cells ×1
• the area factor of e^A is e to the trace of A ×1
• the area factor of the image closes on |det J| as the patch shrinks ×1
• the area multiplier is zero ×1
• the area of the drawn parallelogram is ad − bc ×1
• the array is one the family knows ×1
• the arrow it starts from is a pair of numbers with a length ×1
• the arrow takes at least three steps before it arrives ×1
• the basis is four finite entries no larger than four ×1
• the basis spans an area rather than a line ×1
• the boundary of a triangle's boundary is nothing ×1
• the camera angles are within a half turn ×1
• the census finds a cospectral pair at this size ×1
• the census runs to between four and six points ×1
• the census walks every sign matrix of size 2, 3 or 4 ×1
• the change of basis is invertible ×1
• the classes' orbits cover every labelled graph once ×1
• the column is scaled by between one and three ×1
• the construction is Sylvester's or Paley's ×1
• the correspondence is drawn at size 3, 4 or 5 ×1
• the count holds at every stage of the continuation ×1
• the counting window is between 2 and 8 across ×1
• the crossing pairs are matched exactly by the pairs running to the other finishes ×1
• the curved map is one the family knows ×1
• the determinant is the bound exactly ×1
• the determinant is the number of spanning trees, counted ×1
• the determinant is the product of the lengths times the sine ×1
• the determinant of the product is the product of the determinants ×1
• the determinant over the disc is the degree times the disc's area ×1
• the drawn area is the determinant ×1
• the edges split into what the boundary kills and what it keeps ×1
• the eigenvalues add to the trace ×1
• the eigenvalues add up to the trace ×1
• the eigenvalues of a three-state rate matrix lie within 30° of the negative axis ×1
• the eigenvalues rise in total by exactly t|v|² ×1
• the ellipse has two positive radii no larger than four ×1
• the expansion along a row agrees with the sum over permutations ×1
• the first matrix of a product is four finite entries under 8 in size ×1
• the flat drawing is for the small spaces ×1
• the flow runs for a positive time of at most 4 ×1
• the flow runs for between 0 and 4 units of time ×1
• the flow starts from between three and twelve points ×1
• the flow's own arrival points are the columns of the series' answer ×1
• the form has two real eigenvalues ×1
• the form is positive in every direction ×1
• the four terms of the expansion add to ad − bc ×1
• the graph is one the family knows ×1
• the highest point sits at the first eigen-direction ×1
• the highest value on the circle is the larger eigenvalue of the symmetric part ×1
• the holes in dimension nought are the connected pieces ×1
• the image line has a length to normalise ×1
• the image of the unit square has the determinant's area ×1
• the image turns the way the determinant's sign says ×1
• the incidence matrix's rank is the number of vertices less the number of components ×1
• the integral of the determinant equals the integral of the signed count ×1
• the iteration runs between 2 and 40 steps ×1
• the Laplacian has one zero eigenvalue for each connected piece ×1
• the lattice is drawn far enough out to fill the counting window ×1
• the lattice is drawn out to between 3 and 12 steps ×1
• the line crosses the cusp region ×1
• the logarithm comes out real ×1
• the logarithm exponentiates back to the matrix ×1
• the longest image measured on the ellipse is the first stretch ×1
• the longest radius measured on the curve is 1/√λ for the smaller eigenvalue ×1
• the lowest of the plane-maxima is the middle eigenvalue ×1
• the map does not crush the arrow to nothing at any step ×1
• the map does not flatten the circle to a segment ×1
• the map has two real eigenvalues ×1
• the map has two real eigenvalues to build a basis from ×1
• the map is one of the three the family folds ×1
• the map is one the family knows ×1
• the maps drawn have different ranks ×1
• the marked point is a complex number as a pair ×1
• the matrix drawn here collapses the plane onto a line ×1
• the matrix handed to the symmetric solver is symmetric ×1
• the matrix is four finite entries no larger than four ×1
• the matrix is four finite entries under 8 in size and is not all zero ×1
• the matrix is invertible ×1
• the matrix is square, 2×2 to 6×6, entries under 40 ×1
• the matrix is the adjacency or the Laplacian ×1
• the matrix is three rows of three finite entries no larger than four ×1
• the matrix is three rows of three whole numbers under ten ×1
• the measured Jacobian is the analytic one ×1
• the minmax view is for a 3×3 matrix ×1
• the multiple is a non-zero whole number no larger than four ×1
• the number of independent rows and the number of independent columns are one number ×1
• the numbers are the rationals (0) or the two-element field (2) ×1
• the patch is at most 1.2 across ×1
• the paths counted by hand agree with the binomial ×1
• the permanent of the pattern counts the ways to pick one entry per row and column ×1
• the point is a pair of coordinates ×1
• the points in a window are its area over the determinant, up to the boundary ×1
• the points split into what the boundary kills and what it keeps ×1
• the rate matrix found exponentiates back to the transition matrix ×1
• the rates found reproduce the transition probabilities ×1
• the raw count jumps only where the line crosses the cusp ×1
• the residual falls by four at every halving of the patch ×1
• the right angle reaches the product of the lengths exactly ×1
• the roots add to the trace ×1
• the rotation method and the polynomial agree ×1
• the row operation adds one row to a different one ×1
• the row operation leaves the determinant exactly where it was ×1
• the scaling is one positive weight per row ×1
• the search never beats the divisibility cap ×1
• the search runs to size 3, 4 or 5 ×1
• the second is too ×1
• the series has converged ×1
• the shifted determinant and λ² − (trace)λ + determinant agree ×1
• the sign matrix's determinant is (−2)^(n−1) times the bit matrix's ×1
• the signed count is one all the way along ×1
• the signed count of preimages is the same at every target drawn ×1
• the signed sum is the determinant ×1
• the smallest ellipse holding a lattice point has area at most four times the determinant ×1
• the space is one the family knows ×1
• the starting arrow has a length ×1
• the starting points are between three and twelve pairs of numbers ×1
• the struck row is one of the vertices ×1
• the target is a rotation or minus the identity ×1
• the targets drawn include one-sheet and three-sheet points ×1
• the three columns are not flat, or there is no solid to draw ×1
• the trace of exp L is 2 cos √det L (or cosh), whatever L's entries ×1
• the triangles split into what the boundary kills and what it keeps ×1
• the two directions in the circle are perpendicular ×1
• the two stretches are equal, so every direction is stretched alike ×1
• the two stretches multiply to the area factor ×1
• the unsigned sum is at least as large as the signed one ×1
• the view is one the family draws ×1
• the λ window is a positive width up to 20 ×1
• third ×1
• this transition matrix has exactly two logarithms that are rate matrices, among the branches tried ×1
• three well-separated eigenvalues ×1
• two column lengths between 0.4 and 2 ×1
• two different roots give two independent directions ×1
• two equal columns leave no area at all ×1
• two finishing points, on a small grid ×1
• two of these logarithms do not commute ×1
• two rows agree in exactly half their places ×1
• two starting points, on a small grid ×1
• two to eight matrices ×1
• two to four angles strictly between 5° and 175° ×1
• two to four of the maps the figure knows ×1
• two valid logarithms need the complex eigenvalues to be tiny ×1
• walk lengths up to between 3 and 10 ×1
• what survives and what is lost add to the source's dimension ×1
• what survives and what is lost add to two ×1
• while the raw count takes both one and three ×1

The whole library · What the figures prove