Why does a matrix diagonalise in this case?

  • #1
1,976
1,971
Why does a matrix become diagonal when sandwiched between "change of matrices" whose columns are eigenvectors?
 

Answers and Replies

  • #2
jambaugh
Science Advisor
Insights Author
Gold Member
2,335
313
Change of basis matrices change the basis to the eigen-basis.

In the eigen-basis (the basis formed by the complete set of eigen-vectors).
##M \hat{e}_k = \lambda_k {e}_k##
so ##M## must take the form of a diagonal matrix ##diag(\lambda_1,\lambda_2,\ldots,\lambda_n)##
If you cannot follow that then I suggest you simply do the math for a couple of concrete examples until it clicks.
 
  • #3
mathwonk
Science Advisor
Homework Helper
11,391
1,626
change of basis matrices transform your matrix into one that acts on the columns of the change matrix. since those are eigenvectors, the new matrix is diagonal.
 

Suggested for: Why does a matrix diagonalise in this case?

  • Last Post
Replies
4
Views
507
  • Last Post
Replies
10
Views
521
Replies
5
Views
212
Replies
1
Views
2K
Replies
6
Views
1K
  • Last Post
Replies
9
Views
600
Replies
1
Views
331
  • Last Post
Replies
12
Views
459
  • Last Post
Replies
3
Views
703
  • Last Post
Replies
3
Views
631
Top