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Matrix Diagonalization

Tool to diagonalize a matrix. The diagonalization of a matrix consists of writing it in a base where its elements outside the diagonal are null. This transform was used in linear algebra so that it allow performing simpler calculations.

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Matrix Diagonalization -

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# Matrix Diagonalization

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## Echelon Form Matric Reduction

Tool to diagonalize a matrix. The diagonalization of a matrix consists of writing it in a base where its elements outside the diagonal are null. This transform was used in linear algebra so that it allow performing simpler calculations.

## Answers to Questions

### How to diagonalize a matrix?

A diagonal matrix is a matrix whose elements out of the trace (the main diagonal) are all null (zeros).

A matrix is diagonalizable if there exists an invertable matrix $$P$$ and a diagonal matrix $$D$$ such that $$M = PDP^{-1}$$

To diagonalize a matrix, a method consists in calculating its eigenvectors and its eigenvalues.

Example: The matrix $$M = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$ has for eigenvalues $$3$$ and $$-1$$ and eigenvectors respectively $$\begin{pmatrix} 1 \\ 1 \end{pmatrix}$$ and $$\begin{pmatrix} -1 \\ 1 \end{pmatrix}$$

The diagonal matrix $$D$$ is composed of eigenvalues.

Example: $$D = \begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix}$$

The invertable matrix $$P$$ is composed of the eigenvectors respectively in the same order of the columns.

P must be a normalized matrix

Example: $$P = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$$ Normalization of P: $$P = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$$

### How to prove that a matrix is not diagonalizable?

A matrix is not diagonalizable if it does not have as many dimensions as distinct eigenvectors.

Example: The matrix of dimension 2: $$M = \begin{bmatrix} 5 & 1 \\ 0 & 5 \end{bmatrix}$$ has a double eigenvalue: $$5$$ and therefore a single eigenvector $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ so it is not diagonalizable.

### How to check a diagonalized matrix calculation?

Calculate the inverse of the matrix $$P$$

Check that $$PDP^{-1} = M$$

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