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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...
Eigenvector: stays on its span, only stretches or flips
An eigenvector mathbfv satisfies Amathbfv = lambda mathbfv: the map acts on mathbfv purely by the...
Bridge to the quick $2\times2$ eigenvalue trick
Chapter 15 compresses the quadratic characteristic polynomial for 2 times 2 matrices using trace...
The naive cross-interchange identity usually fails
For a general linear map T:mathbbR^3tomathbbR^3, do not assume T(mathbfatimesmathbfb)=Tmathbfatimes...
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...
Algebraic cofactor mirror
Minors and cofactors reorganize the same volume ratios into Laplace expansion form [@2]. The...
Mean, square-root mnemonic
If two numbers share mean m and product p, they sit symmetrically about m: write them as mpm d for...
Many things are "vectorish"
Arrows in the plane, tuples of numbers, polynomials, and signals can all reuse linear algebra...
Why depth needs nonlinear breaks
Before the algebra, the narrative asks for a hope about hidden layers: maybe a penultimate layer...
Stochasticity: minibatches approximate the full-data gradient
Exact gradient descent recomputes nabla C using every training example each step. For MNIST that is...
Momentum and adaptive methods (conceptual)
Plain SGD uses only the current batch gradient. Momentum maintains a velocity vector mathbfv that...