Is this matrix diagonalizable?

Thread starter scholesmu
Start date Apr 13, 2010
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Matrix

In summary, a diagonalizable matrix is a square matrix that can be transformed into a diagonal matrix through a similarity transformation. This is possible if the matrix has n distinct eigenvalues, where n is the size of the matrix. A diagonalizable matrix has many useful properties and is easier to work with, but it must be a square matrix. However, not all diagonalizable matrices are invertible, as long as at least one eigenvalue is equal to zero.

Apr 13, 2010

scholesmu

A= (3 0
0 3 )

why is it diagonalizable?
i'm not sure...

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Apr 13, 2010

Rajini

Your matrix is already diagonal-> so i guess it can be diagonalized.

1. What does it mean for a matrix to be diagonalizable?

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 represented as a diagonal matrix with all zero entries except for the main diagonal.

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

A matrix is diagonalizable if it has n distinct eigenvalues, where n is the size of the matrix. This can be determined by finding the eigenvalues and eigenvectors of the matrix and checking if they are linearly independent.

3. What is the significance of a matrix being diagonalizable?

A diagonalizable matrix is easier to work with and has many useful properties, such as being easier to compute powers and inverses of the matrix. It also allows for simpler analysis of the matrix's behavior and relationships with other matrices.

4. Can a non-square matrix be diagonalizable?

No, only square matrices can be diagonalizable since the similarity transformation requires the same number of rows and columns.

5. Is every diagonalizable matrix invertible?

Not necessarily. A matrix can be diagonalizable but not invertible, as long as it has at least one eigenvalue equal to zero. In this case, the matrix will have a zero determinant and will not have an inverse.