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Characteristic polynomial: $\det(A-\lambda I)=0$
Search for scalars lambda such that A - lambda I is singular. A nontrivial kernel then supplies...
Eigenbasis and diagonalization
When eigenvectors form a basis, A = B D B^-1 with diagonal D listing eigenvalues on the diagonal...
Trace and determinant from eigenvalues (with multiplicity)
Over mathbbC, counting algebraic multiplicities, mathrmtr(A) = sum_i lambda_i, qquad det(A) =...
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...
Trace and determinant encode eigenvalue sum and product
For a 2times 2 matrix A=beginbmatrixa&bc&dendbmatrix, the characteristic polynomial is...
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...
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]....
Series finale prelude: abstraction ahead
Chapter 16 steps back from coordinates to the idea of a vector space as a set with addition and...
Many things are "vectorish"
Arrows in the plane, tuples of numbers, polynomials, and signals can all reuse linear algebra...
Linearity revisited: additivity + homogeneity
A map T is linear when T(u+v) = T(u) + T(v) and T(cu) = cT(u). That pair of rules generalizes grid...