Conditions for diagonalizable matrix

So take one of the eigenvectors and put it in your formula.In summary, when a 3×3 matrix A has 3 linearly independent eigenvectors, it can be written as a matrix P multiplied by its inverse, and then multiplied by A to produce a diagonal matrix D. To prove that different eigenvalues produce linearly independent eigenvectors, we can use the equations A*x = c1x and A*y = c2y, where c1 and c2 are different eigenvalues. If we assume that a1x + a2y = 0, then we can show that a1 = a2 = 0, proving that the eigenvectors are linearly independent. This is because if
  • #1
Kaguro
221
57
Homework Statement
Show that if a matrix has n distinct eigenvalues then it is diagonalizable.
Relevant Equations
A matrix is diagonalizable if it is similar to a diagonal matrix.
If a 3×3 matrix A produces 3 linearly independent eigenvectors then we can write them columnwise in a matrix P(non singular). Then the matrix D = P_inv*A*P is diagonal.

Now for this I need to show that different eigenvalues of a matrix produce linearly independent eigenvectors.

A*x = c1x
A*y = c2y

c1 !=c2

Then how to show that :

a1x + a2y =0 implies a1=a2=0?
 
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  • #2
If you have ##n## different eigenvalues, what can you say about the characteristic polynomial and the kernel?
 
  • #3
I assume you mean to say an nxn matrix with n distinct eigenvalues.
 
  • #4
Use c1x+c2y=0. Both most be on the same line. Lines deprend on a single parameter, meaning lines through the origin.
 

FAQ: Conditions for diagonalizable matrix

What is a diagonalizable matrix?

A diagonalizable matrix is a square matrix that can be transformed into a diagonal matrix through a similarity transformation. This means that the matrix can be expressed as a product of three matrices: A = PDP-1, where P is an invertible matrix and D is a diagonal matrix.

What are the conditions for a matrix to be diagonalizable?

The conditions for a matrix to be diagonalizable are:

  • The matrix must be a square matrix.
  • The matrix must have n linearly independent eigenvectors, where n is the size of the matrix.
  • The eigenvectors must span the entire vector space of the matrix.
  • The matrix must have distinct eigenvalues.

How can I determine if a matrix is diagonalizable?

To determine if a matrix is diagonalizable, you can follow these steps:

  • Find the eigenvalues of the matrix.
  • For each eigenvalue, find its corresponding eigenvectors.
  • If the matrix has n distinct eigenvalues and n linearly independent eigenvectors, then it is diagonalizable.

Can a non-square matrix be diagonalizable?

No, a non-square matrix cannot be diagonalizable. Diagonalizable matrices must be square matrices, meaning they have the same number of rows and columns.

What is the significance of diagonalizable matrices?

Diagonalizable matrices are important in linear algebra because they are easier to work with and understand. They can also simplify calculations and make it easier to find solutions to certain problems. Additionally, diagonalizable matrices have many applications in fields such as physics, engineering, and economics.

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