I wonder if the matrices are similar, and if this is true, then if I want to solve a minimal polynomial problem on a matrix A, if I can simplify the matrix A until it has the form of a diagonal matrix by blocks B.
121640
I am trying to understand reduction of an unsymmetric real square matrix to Hessenberg form from Numerical Recipes Vol. 3.  In it, the author states that one does not use Gaussian elimination for reducing to Hessenberg form because the Gaussian elimination does not preserve similarity and hence ends up changing the eigenvalues of the matrix, which is undesirable.  Why does Gaussian elimination not preserve similarity?   Also, what conditions decide whether a particular matrix transformation preserves similarity?  Heres a  to where I found it. In case you cant view that, it's on page 594 of Numerical Recipes in C++ (Volume 3).
116545
Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant, trace and characteristic polynomial, but that are not similar to each other.   I come up with two matrices: $A=\begin{pmatrix} 0 &amp;1 \\  0 &amp;0  \end{pmatrix}$ and $B=\begin{pmatrix} 0 &amp;0 \\  1 &amp;0  \end{pmatrix}.$ It is easy to check that they have same rank, determinant, trace and characteristic polynomial. However, my question is I do not know how to prove two matrices are similar or not. I have learnt the converse in my textbook, i.e. If two matrices are similar, they have the same determinant, characteristic polynomial,etc. I have also known that (but I do now know the proof), we can check by using Jordan form of two matrices. I do not know if this claim is correct: &quot;If two matrices have the same Jordan form, they are similar to each other.&quot; Yet, go back to the question, is it a quick way to prove? Thank you in advance.
20860
I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester.  I took the polynomial to mean that there were at least two 2's and one 1 on the main diagonal, and that the largest Jordan block with eigenvalue 2 is $2\times 2$ while the largest Jordan block with eigenvalue 1 is $1\times 1$.  Did I miss any matrices or interrupt the minimal polynomial incorrectly?  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;2  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;2  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;2  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;2  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;2  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;1  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;1  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}  \begin{pmatrix}   2 &amp;0  &amp;0  &amp;0  &amp;0\\   0 &amp;2  &amp;0  &amp;0  &amp;0\\   0 &amp;0  &amp;1  &amp;0  &amp;0\\   0 &amp;0  &amp;0  &amp;1  &amp;0\\   0 &amp;0  &amp;0  &amp;0  &amp;1 \end{pmatrix}
34722
It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true?     If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?   Edit: Addionally, what happens in $M_n(\mathbb{R})$?  Nota Bene: As a corollary of the counterexamples below, we deduce that if $A$ is not diagonal then $e^A$ may be diagonal.
86124
Currently I'm stuck with this problem:     The following matrices are similar to each other.      $$A=\begin{bmatrix}-3&amp;0\\8&amp;4\end{bmatrix},\qquad B=\begin{bmatrix}-4&amp;4\\-2&amp;5\end{bmatrix},\qquad C=\begin{bmatrix}k&amp;1\\m&amp;-2\end{bmatrix}.$$      How do you find the matrix $P$ such that $AP = PB$?       And how can you calculate the values of $k$ and $m$?      Thank you in advance.
3591
I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is neither diagonal nor triangular.  Is there a term for such matrices, and have they been researched?
37567
A real $2 \times 2 $ matrix $M$ such that $$M^2 =  \begin{pmatrix} -1&amp;0 \\ 0&amp;-1-\epsilon  \\   \end{pmatrix}$$ (a) exists for all $\epsilon &gt; 0$. (b) does not exist for any $\epsilon &gt; 0$. (c) exists for some $\epsilon &gt; 0$. (d) None of the above is true.  Attempt: I couldn't think of any theory base to prove / disprove the existence of such a matrix. Could someone please give me a hint on how to go about this problem? Thank you very much for your help in this regard.
18833
Let $A$ be a complex matrix of size $n$ if for some positive integer $k$ , $A^k=I_n$ , then is $A$ diagonalizable ?
102543
I have a problem with this. Actually, still don't have the right way to start :/  Problem :  Let $A$ and $B$ be $n \times n$ complex matrices such that $AB - B^2A^2 = I_n$. Prove that if $A^3 + B^3 = 0$, then $BA - A^2B^2 = I_n$.  Thanks for any help.
8479
Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then they are similar in $K^{n\times n}$?   Any help ... thanks!
27490
The below statement is a true/false exercise.  Statement: For all square matrices A and B of the same size, it is true that $(A + B)2 = A^2 + 2AB + B^2$.  My thought process: Since it is not a proof, I figure I can show by example and come to a valid conclusion based on such example.  My work:  Come up with a square matrix A and B let both be a 2 by 2 matrix(rows and cols must be same).  Matrix $A$:  $A = \begin{array}{ccc} 3 &amp; 5 \\ 4 &amp; 6 \\ \end{array} $  Matrix $B$:  $B =  \begin{array}{ccc} 5 &amp; 8 \\ 9 &amp; 4 \\ \end{array} $  $A + B =  \begin{array}{ccc} 8 &amp; 13 \\ 13 &amp; 10 \\ \end{array}$  $(A + B)^2 =  \begin{array}{ccc} 233 &amp; 234 \\ 234 &amp; 264 \\ \end{array}$  $A^2 =  \begin{array}{ccc} 29 &amp; 45 \\ 36 &amp; 56 \\ \end{array}$  $(AB) =  \begin{array}{ccc} 60 &amp; 44 \\ 74 &amp; 56 \\ \end{array}$  $2(AB) =  \begin{array}{ccc} 120 &amp; 88 \\ 234 &amp; 112 \\ \end{array}$  $B^2 =  \begin{array}{ccc} 97 &amp; 72 \\ 81 &amp; 88 \\ \end{array}$  $A^2 + 2AB + B^2 =  \begin{array}{ccc} 246 &amp; 205 \\ 265 &amp; 256 \\ \end{array}$  Based my above work, the answer is false.   Is there another way to approach the problem? It seems like a lot of work needed to be done for a true/false question which raised my suspicion about whether there is a better way to look at the problem.
76337
Like the title says, can a matrix in $\mathbb{R}$ have a minimal polynomial that has coefficients in $\mathbb{C}$?  I have a feeling the answer is yes because the characteristic polynomial can have coefficients in $\mathbb{C}$ and so I don't see any reason why not, I was merely asking in case there was some sneaky proof that proves that this cannot happen.
101417
Can anyone help me to understand what is suppose to do in this question?  "A matrix that is similar to the identity matrix" I should say something about this but I do not understand what is meant to do.  I know that two n x n matrices A and B are similar if $B=P^{-1}AP $.   They also have some properties such as:  The same:  Rank; Characteristic equation;  Determinant; Trace; Eigenvalues etc.   However I do not understand the question and consequently I do not know how to start.  Can anyone help me on this?  Thanks
41104
I have to show that all $n \times n$ nilpotent matrices of order n are similar. My initial approach was to show that for all nilpotent matrices their minimun characteristic polynomial is of the form: $$\lambda^n$$  Is this sufficient?  Can someone show me a formal approach to this problem?  Thanks!
61110
I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix.   What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.  Also, any suggested texts which have good coverage on this topic would be very helpful.
94166
Can someone show me step-by-step how to diagonalize this matrix? I'm trying to teach myself differential equations + linear algebra, but I'm stumped on how to do this. I'd really appreciate if someone would take the time to do this with me!   $\begin{bmatrix} 1 &amp; 2 &amp; 0 \\ 2 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; -3 \\ \end{bmatrix}$
13677
Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial?    I think the easy case is when the operator has $n$ distinct eigenvalues, but what about if it is diagonalizable? Is that sufficient, or can there be cases (with repeated eigvals) when char poly doesn't equal min poly? What are the general conditions when the equality holds? Is it possible to define them without use of determinant? (I am working by Axler and he doesn't like it.)  Thanks.