[Verse 1]
Matrix A sits waiting, vectors dance around
But some directions hold their ground
When A transforms, these special few
Keep pointing where they always knew
[Pre-Chorus]
A times v equals lambda v
The eigenvector stays so free
Only stretches, never bends
Along the axis where it tends
[Chorus]
Eigenvalues, eigenvectors too
The grain of wood that splits right through
Find the paths that never turn
Characteristic equations we must learn
Det of A minus lambda I
Set to zero, that's the why
Diagonalize and simplify
The matrix grain will multiply
[Verse 2]
Subtract lambda from the main diagonal line
Determinant zero is our sign
Polynomial roots reveal the scale
Each eigenvalue tells the tale
[Pre-Chorus]
Sum of roots equals the trace
Product gives determinant's place
N complex roots we always find
Thanks to algebra's grand design
[Chorus]
Eigenvalues, eigenvectors too
The grain of wood that splits right through
Find the paths that never turn
Characteristic equations we must learn
Det of A minus lambda I
Set to zero, that's the why
Diagonalize and simplify
The matrix grain will multiply
[Bridge]
P D P inverse is the key
Columns hold the vectors free
When geometric meets algebraic
Diagonalization works like magic
Cayley-Hamilton speaks the truth
Every matrix satisfies proof
Power k becomes so clean
In the eigenbasis scene
[Chorus]
Eigenvalues, eigenvectors too
The grain of wood that splits right through
Find the paths that never turn
Characteristic equations we must learn
Det of A minus lambda I
Set to zero, that's the why
Diagonalize and simplify
The matrix grain will multiply
[Outro]
Matrix grain runs straight and true
Eigenvectors guide you through
Linear independence makes it right
Diagonalization brings the light