[Verse 1]
Matrix A with two rows calling, find what makes it special
Two by one above the zero, three sits on diagonal
Subtract lambda from the main line, determinant equation
Lambda squared minus five lambda plus six gives salvation
[Chorus]
Eigenvalues dancing, two and three appear
Eigenvectors follow where the nullspace makes it clear
Diagonalizable treasure when we have enough
Linear independence proves the matrix has the stuff
[Verse 2]
If A's invertible and mighty, zero can't be eigenspace
Ax equals zero times x means only zero shows its face
But invertible means nullspace shrinks to just the origin
Non-zero eigenvectors can't exist where zero's sovereign
[Chorus]
Eigenvalues dancing, two and three appear
Eigenvectors follow where the nullspace makes it clear
Diagonalizable treasure when we have enough
Linear independence proves the matrix has the stuff
[Verse 3]
Second derivative plus y equals zero, differential calling
Sine and cosine are the heroes, never rising, never falling
Two dimensions span solutions, basis vectors oscillating
Linear combinations weaving, harmonic patterns generating
[Bridge]
Rank plus nullity equals n, dimensions never lie
Gram-Schmidt takes three vectors, orthogonal they fly
Complex eigenvalues travel with their conjugate twin
SVD splits matrices where singular values begin
[Verse 4]
Cauchy-Schwarz inequality, expand the perfect square
Discriminant is non-negative, proving bounds with mathematical flair
Inner products hold the secret, dot product magnitude
Triangle inequality follows from this algebraic attitude
[Outro]
From eigenspaces to orthogonal bases
Linear algebra reveals its traces
Theorems proven, structures shown
Mathematical kingdom we now own