Why does a matrix diagonalise in this case?

In summary, when a matrix is sandwiched between "change of matrices" whose columns are eigenvectors, it becomes diagonal. This is because the change of basis matrices transform the original matrix into one that acts on the columns of the change matrix, resulting in a diagonal matrix in the eigen-basis. This can be understood by doing the math for concrete examples.
  • #1
Wrichik Basu
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Why does a matrix become diagonal when sandwiched between "change of matrices" whose columns are eigenvectors?
 
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  • #2
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
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.
 

1. Why is it important for a matrix to be diagonalizable?

Diagonalizable matrices have a simpler structure and are easier to work with in calculations. They also have special properties that make them useful in various applications, such as optimization problems and differential equations.

2. How do I know if a matrix is diagonalizable?

A square matrix is diagonalizable if it has n linearly independent eigenvectors, where n is the size of the matrix. This means that the matrix can be written as a diagonal matrix, with the eigenvalues of the matrix on the main diagonal.

3. Can every matrix be diagonalized?

No, not every matrix can be diagonalized. A matrix can only be diagonalized if it has n linearly independent eigenvectors, where n is the size of the matrix. This means that not all matrices have a simpler diagonal form.

4. What is the relationship between diagonalization and eigenvalues?

Diagonalization is the process of finding a diagonal matrix that is similar to the original matrix. This diagonal matrix has the eigenvalues of the original matrix on the main diagonal. In other words, the eigenvalues are the entries on the main diagonal of the diagonalized matrix.

5. How does diagonalization help solve systems of linear equations?

Diagonalization simplifies systems of linear equations by transforming them into a diagonal form. This makes it easier to solve for the unknown variables and find the solutions to the system. This is especially useful for large systems of equations, as diagonalization reduces the number of calculations needed.

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