Is Matrix A Diagonalizable? Exploring the Proof and Implications

Thread starter pezola
Start date Mar 21, 2008

In summary, a matrix is diagonalizable if it can be transformed into a diagonal matrix through a similarity transformation. To prove diagonalizability, a complete set of eigenvectors must be found. There are two conditions for a matrix to be diagonalizable and a non-square matrix cannot be diagonalizable. Diagonalizability is closely related to eigenvalues and eigenvectors, as a matrix is diagonalizable if it has a complete set of eigenvectors and the eigenvalues appear on the diagonal of the diagonal matrix.

Mar 21, 2008

pezola

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Mar 21, 2008

tiny-tim

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mmm … gimme a clue! …

Mar 21, 2008

pezola

yeah, i know... i just submitted the real one...can I delete this blank one?

Mar 21, 2008

tiny-tim

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I don't know …

Let's leave a trail for people to follow: :bugeye:

https://www.physicsforums.com/showthread.php?t=223399

What does it mean for a matrix to be diagonalizable?

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

How do you prove that a matrix A is diagonalizable?

To prove that a matrix A is diagonalizable, we need to show that it has a complete set of eigenvectors. This means that we need to find a set of linearly independent eigenvectors that span the vector space, and these eigenvectors will form the columns of the matrix P in the similarity transformation A = PDP^-1.

What are the conditions for a matrix A to be diagonalizable?

There are two conditions for a matrix A to be diagonalizable: 1. A must have n linearly independent eigenvectors, where n is the dimension of the matrix. 2. The sum of the dimensions of the eigenspaces corresponding to each distinct eigenvalue must equal the dimension of the matrix.

Can a non-square matrix be diagonalizable?

No, a non-square matrix cannot be diagonalizable. This is because a diagonal matrix must have the same number of rows and columns, and a non-square matrix does not have a diagonal form.

How is diagonalizability related to eigenvalues and eigenvectors?

Diagonalizability is closely related to eigenvalues and eigenvectors. A matrix is diagonalizable if and only if it has a complete set of eigenvectors, and these eigenvectors form the columns of the matrix P in the similarity transformation A = PDP^-1. The eigenvalues of the matrix A will be the entries on the diagonal of the diagonal matrix D.