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Relation to the quadratic formula
Normalize the characteristic polynomial to x^2-sx+p with root sum s and root product p. Vieta says...
Limitations beyond $2\times 2$
The shortcut is deliberately narrow: it is for 2times 2 eigenvalues you want quickly on paper [@1]....
Composition: the matrix nearest the vector acts first
If S applies first then T, the composite on column vectors is TS, so (TS)mathbfx = T(Smathbfx)...
Non-commutativity: order usually matters
Even when both AB and BA exist, they often differ. Different pipelines deform space differently...
Trace and determinant stay similar
Similarity B^-1MB preserves trace and determinant even when individual entries scramble [@2]. That...
Non-invertible maps collapse at least one direction
Dependent columns mean several inputs share outputs; the kernel captures those differences....
Packing the chapter for matrix multiplication next time
The narrative ends poised for composition: you now read Amathbfx as geometry encoded in columns....
Row picture: rows of $A$ dot columns of $B$
Entry (i,j) of AB is row i of A dotted with column j of B. Useful when interpreting rows as...
Characteristic polynomial: $\det(A-\lambda I)=0$
Search for scalars lambda such that A - lambda I is singular. A nontrivial kernel then supplies...
Trace and determinant encode eigenvalue sum and product
For a 2times 2 matrix A=beginbmatrixa&bc&dendbmatrix, the characteristic polynomial is...
Coordinates as linear combinations of basis arrows
Writing (x,y) as xhatmathbfi+yhatmathbfj reveals the secret grammar: coordinates list the scalars...
Chapter close: language hygiene before transformations
The narrative tightens vocabulary: distinguish vectors , points , coordinates , and bases before...