Diagonalization

Dependencies:

1. Eigenvalues and Eigenvectors
2. Product of stacked matrices

Let $A$ be an $n$ by $n$ matrix. Let $P$ be an $n$ by $k$ matrix. Let $v_i$ be the $i^{\textrm{th}}$ column of $P$. Let $D$ be a $k$ by $k$ diagonal matrix where $D[i, i] = d_i$.

Then $AP = PD \iff \forall i, (d_i, v_i)$ is an eigenvalue-eigenvector pair.

If $P$ is an invertible square matrix, this means that $D = P^{-1}AP$. $A$ is said to be diagonalizable iff $\exists$ an invertible matrix $P$, such that $P^{-1}AP$ is diagonal.

Proof

\[ (PD)[i, j] = \sum_{t=1}^k P[i, t]D[t, j] = P[i, j]D[j, j] = (v_j)_i d_j = (d_jv_j)_i \]

\[ (AP)[i, j] = (A[v_1|v_2|\ldots|v_k])[i, j] = [Av_1|Av_2|\ldots|Av_k][i, j] = (Av_j)_i \]

\begin{align} & AP = PD \\ &\iff \forall i, \forall j, (AP)[i, j] = (PD)[i, j] \\ &\iff \forall i, \forall j, (Av_j)_i = (d_jv_j)_i \\ &\iff \forall j, Av_j = d_jv_j \\ &\iff \forall j, (d_j, v_j) \textrm{ is an eigenvalue-eigenvector pair} \end{align}

Dependency for:

1. A is diagonalizable iff there are n linearly independent eigenvectors
2. Orthogonally diagonalizable iff hermitian

Info:

• Depth: 12
• Number of transitive dependencies: 49

Transitive dependencies:

1. /linear-algebra/matrices/gauss-jordan-algo
2. /linear-algebra/vector-spaces/condition-for-subspace
3. /sets-and-relations/equivalence-relation
4. /sets-and-relations/composition-of-bijections-is-a-bijection
5. Group
6. Ring
7. Polynomial
8. Field
9. Vector Space
10. Linear independence
11. Span
12. Linear transformation
13. Composition of linear transformations
14. Vector space isomorphism is an equivalence relation
15. Integral Domain
16. Comparing coefficients of a polynomial with disjoint variables
17. Semiring
18. Matrix
19. Stacking
20. System of linear equations
21. Product of stacked matrices
22. Matrix multiplication is associative
23. Reduced Row Echelon Form (RREF)
24. Elementary row operation
25. Every elementary row operation has a unique inverse
26. Row equivalence of matrices
27. Matrices over a field form a vector space
28. Row space
29. Row equivalent matrices have the same row space
30. RREF is unique
31. Identity matrix
32. Inverse of a matrix
33. Inverse of product
34. Elementary row operation is matrix pre-multiplication
35. Row equivalence matrix
36. Equations with row equivalent matrices have the same solution set
37. Basis of a vector space
38. Linearly independent set is not bigger than a span
39. Homogeneous linear equations with more variables than equations
40. Rank of a homogenous system of linear equations
41. Rank of a matrix
42. Basis of F^n
43. Matrix of linear transformation
44. Coordinatization over a basis
45. Basis changer
46. Basis change is an isomorphic linear transformation
47. Vector spaces are isomorphic iff their dimensions are same
48. Canonical decomposition of a linear transformation
49. Eigenvalues and Eigenvectors