LA 101
M04 · L03
Module 4: Eigenvalues

Diagonalization

Write A = PDP⁻¹ where D is diagonal and P holds the eigenvectors. This unlocks matrix powers, exponentials, and differential equations in one move.

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LA 101
M04 · L03
The Decomposition

A = PDP⁻¹

P's columns are eigenvectors; D's diagonal entries are the eigenvalues. P must be invertible — the eigenvectors must be linearly independent.

Eigendecomposition
A=PDP^{-1},\quad D=\text{diag}(\lambda_1,\ldots,\lambda_n)
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LA 101
M04 · L03
When Is A Diagonalizable?

n Independent Eigenvectors

  • A is diagonalizable iff it has n linearly independent eigenvectors
  • Sufficient: n distinct eigenvalues → always diagonalizable
  • Repeated eigenvalue OK if eigenspace dimension = algebraic multiplicity
  • Defective = not diagonalizable → need Jordan form
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LA 101
M04 · L03
The Procedure

Five Steps to Diagonalize

  • Find all eigenvalues from det(A − λI) = 0
  • Find n linearly independent eigenvectors
  • Form P: eigenvectors as columns
  • Form D: eigenvalues on diagonal, same order
  • Verify: AP = PD
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LA 101
M04 · L03
Example: A = [[4,1],[2,3]]

Building P and D

Eigenvalues
λ₁ = 5 → v₁ = [1, 1]ᵀ  |  λ₂ = 2 → v₂ = [−1, 2]ᵀ
Factorization
\begin{bmatrix}4&1\\2&3\end{bmatrix}=\begin{bmatrix}1&-1\\1&2\end{bmatrix}\begin{bmatrix}5&0\\0&2\end{bmatrix}\begin{bmatrix}1&-1\\1&2\end{bmatrix}^{-1}
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LA 101
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The Big Payoff

Matrix Powers in One Step

A = PDP⁻¹ gives Aᵏ = PDᵏP⁻¹. Computing Dᵏ just means raising each diagonal scalar to the k-th power — no matrix chains needed.

Power Formula
A^k=PD^kP^{-1}=P\,\text{diag}(\lambda_1^k,\ldots,\lambda_n^k)\,P^{-1}
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LA 101
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Geometric Picture

Pure Scaling in the Eigenbasis

In the eigenvector coordinate system, A just scales each axis independently. P⁻¹ converts to that basis, D scales, P converts back. No mixing between directions.

P⁻¹
To eigenbasis
D
Scale axes
P
Back to standard
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LA 101
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Spectral Theorem

Symmetric → Orthogonal P

Real symmetric matrices (Aᵀ = A) are always diagonalizable with an orthogonal P: A = QDQᵀ where QᵀQ = I.

  • All eigenvalues are real
  • Eigenvectors from different eigenspaces are orthogonal
  • P can be chosen orthonormal: Qᵀ = Q⁻¹ (cheap to invert)
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LA 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

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LA 101
M04 · L03
Application: ODEs

Matrix Exponential

For x'(t) = Ax, the solution is x(t) = e^{At}x(0). Diagonalization gives e^{At} = Pe^{Dt}P⁻¹ — n scalar exponentials, not a matrix series.

Matrix Exponential
e^{At}=P\,\text{diag}(e^{\lambda_1 t},\ldots,e^{\lambda_n t})\,P^{-1}
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LA 101
Up Next
Lesson 3 Complete

M4-L4: Applications

You can now diagonalize a matrix. Next: see eigenvalues in action — Google's PageRank, PCA, Markov chains, and structural vibration analysis.

Coming Up
M4-L4: Applications of Eigenvalues →
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