A is n by n matrix. It is diagonalized by P. Find the matrix that

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In summary, the matrix that diagonalizes the transpose of a n by n matrix A is (P^t)^-1, where P is the matrix that diagonalizes A.
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charlies1902
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A is n by n matrix. It is diagonalized by P. Find the matrix that diagonalizes the tranpose of A.


We have the equation A=PDP^-1
Where D is the diagonal matrix consisting of the eigen values of A.


(A)^t=(PDP^-1)^t
A^t=(P^-1)^t * D^t * P^t
A^t=(P^t)^-1 * D * P^t

So the matrix (P^t)^-1 diagonalizes A^t.

Is that correct?
 
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charlies1902 said:
A is n by n matrix. It is diagonalized by P. Find the matrix that diagonalizes the tranpose of A.


We have the equation A=PDP^-1
Where D is the diagonal matrix consisting of the eigen values of A.


(A)^t=(PDP^-1)^t
A^t=(P^-1)^t * D^t * P^t
A^t=(P^t)^-1 * D * P^t

So the matrix (P^t)^-1 diagonalizes A^t.

Is that correct?

Sounds fine to me.
 

FAQ: A is n by n matrix. It is diagonalized by P. Find the matrix that

What is a diagonal matrix?

A diagonal matrix is a square matrix in which all the elements outside of the main diagonal (the diagonal from the upper left to the lower right) are equal to zero. This means that the only non-zero elements in the matrix are on the main diagonal.

How is a matrix diagonalized?

A matrix is diagonalized by finding a matrix P that, when multiplied by the original matrix A, results in a diagonal matrix D. This is done through a process called diagonalization, which involves finding the eigenvalues and eigenvectors of the original matrix.

What is the significance of diagonalizing a matrix?

Diagonalizing a matrix can simplify certain calculations and make it easier to understand the properties of the matrix. It also allows for easier computation of powers and inverses of the matrix.

How do you find the matrix P that diagonalizes A?

The matrix P can be found by using the eigenvectors of A as the columns of P. These eigenvectors must be linearly independent and the eigenvalues must correspond to the columns in the same order as the eigenvectors.

Can any matrix be diagonalized?

Not all matrices can be diagonalized. For a matrix to be diagonalizable, it must have a complete set of linearly independent eigenvectors. If the matrix does not have enough linearly independent eigenvectors, it cannot be diagonalized.

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