So when dealing with a linear transformation, after we have computed the matrix of the linear transformation, and we are asked "is this matrix diagonalizable", I begin by finding the eigenvalues and eigenvectors using the characteristic equation.(adsbygoogle = window.adsbygoogle || []).push({});

Once I have found eigenvectors, if I see these vectors are linearly independent, then a theorem tells us the matrix of the linear transformation is, in fact, diagonalizable. Another theorem tells us that this matrix, D, has entries along its main diagonal that are the eigenvalues of the matrix of the linear transformation.

My questions are as follows: what about the order of these eigenvalues? When I solve the characteristic equation, I can order the roots however I want. This means that the matrix of the transformation is diagonalizable to up to n! different diagonal matrices (actually, exactly n! different matrices if we consider complex roots, since all polynomials of degree n contain exactly n roots). My other question is: when choosing eigenvectors that correspond to eigenvalues, can we just choose any vector in the eigenspace, since often times, our eigenvector contains arbitrary constants that can be any real number? I will illustrate my question with an example:

For the matrix A =

[itex]\left[ \begin{array}{ccc}

1 & 1 \\

-2 & 4 \end{array} \right][/itex]

Solving for the characteristic polynomial, we have

[itex]\lambda = 2, \lambda = 3[/itex]

whose associated eigenvectors are

[itex] \left[ \begin{array}{ccc}

1 \\

1 \end{array} \right][/itex]

and

[itex]\left[ \begin{array}{ccc}

1 \\

2 \end{array} \right][/itex]. These eigenvectors were chosen with a random scalar constant; can we choose any other eigenvectors?

These two vectors are linearly independent, thus the matrix A can be diagonalized. We see that matrix A is similar to matrix D, where D =

[itex]\left[ \begin{array}{ccc}

2 & 0 \\

0 & 3 \end{array} \right][/itex].

Could we not have stated the roots in a different order, and then eventually conclude that A is similar to D, where D =

[itex]\left[ \begin{array}{ccc}

3 & 0 \\

0 & 2 \end{array} \right][/itex]?

I'm confused as to what order we are to choose, if it even matters (which it seems it does. Sorry if this is a "homework type question", but I think my question is more general and not a specific textbook style question. Sorry if my question is confusing. Any help is appreciated.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Diagonalization and similar matrices

Loading...

Similar Threads for Diagonalization similar matrices | Date |
---|---|

I Diagonalization and change of basis | Jan 16, 2018 |

B Why does a matrix diagonalise in this case? | Nov 21, 2017 |

A reasonable analogy for understanding similar matrices? | Dec 12, 2015 |

Similar Diagonal Matrices | Nov 6, 2012 |

Help! Diagonal matrix similar to upper triangular matrix? | Dec 16, 2007 |

**Physics Forums - The Fusion of Science and Community**