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Trace and determinant from eigenvalues (with multiplicity)
Over mathbbC, counting algebraic multiplicities, mathrmtr(A) = sum_i lambda_i, qquad det(A) =...
Characteristic polynomial: $\det(A-\lambda I)=0$
Search for scalars lambda such that A - lambda I is singular. A nontrivial kernel then supplies...
Geometry: eigen directions reveal intrinsic stretching
Reading columns shows where basis vectors land, but eigenvectors summarize natural axes of strain...
Bridge to the quick $2\times2$ eigenvalue trick
Chapter 15 compresses the quadratic characteristic polynomial for 2 times 2 matrices using trace...
Example walkthrough without carrying minors
The workflow is recite, then compute: read t=a+d and p=det A, set m=t/2, then write...
Same words, new habitats: eigenfunctions, inner products
Eigenvectors become eigenfunctions when the vector space is a function space. Dots become inner...
Grids become parallelepiped lattices, still linear
Chapter 4 made two-dimensional grids bend into parallelogram lattices while preserving the linear...
Linear combinations: scale, then add, same recipe in every dimension
A linear combination of vectors mathbfv_1,ldots,mathbfv_k is any sum...
Rank in three dimensions: lines, planes, volumes
Rank counts how many independent directions survive in the output. Rank 1 collapses all of...
Linear independence: no hidden redundancy in the generator list
Vectors are linearly independent when the only combination giving mathbf0 is the trivial one with...
Geometric story before cofactor expansions
Pictures come before Laplace expansions here [@1]. Understand collapse events first: projections...
Chapter synthesis: keep both absolute value and sign in mind
Chapter synthesis: lvertdet Arvert answers how much volume scales; operatornamesign(det A) answers...