For questions about matrix diagonalization, that is, writing a matrix, a bilinear form or an operator into a "basis" making this one diagonal. This tag is **NOT** for diagonalization arguments from logic and set theory.

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2
votes
1answer
30 views

Diagonalizing the X and Z matrices

I've got two special matrices I'm trying to diagonalize : The Z matrix :$$\begin{bmatrix} 1&1&\cdots&1&1\ \\&&&1 \\&&\diagup \\&1 ...
4
votes
1answer
40 views

Basis of the matrices with only non diagonalizable matrices

Is it possible to find a basis of $M_n(\mathbb{R})$ that only has non diagonalisable matrices ? I'm looking for a rather easy example, or a proof of the (non-)existence.
-3
votes
1answer
13 views

Let A be a 2*2 matrix on R. Prove that if (trace(A))^{2} > 4Det(A), then A is diagonalizable over R. [on hold]

Let A be a 2*2 matrix on R. Prove that if (trace(A))^{2} > 4Det(A), then A is diagonalizable over R.
3
votes
2answers
31 views

Linear algebra: diagonalisation of antisymmetrisation

I'm facing an apparent contradiction when trying to solve a linear algebra exercise. I am asked to find a basis for the vector space of $2\times 2$ matrices such that the function $$f(A) = A - A^t $$ ...
1
vote
0answers
29 views

Eigenvalues with constraints?

Note: This is a short version of About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms) For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists ...
1
vote
3answers
48 views

Prove that a matrix with a given characteristic polynomial is diagonalizable

Matrix $A$ is defined over real number. Characteristic polynomial : $p(x)=(x+3)^2(x-1)(x-5)$ It also known that : $$\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$$ prove $A$ ...
1
vote
0answers
40 views

Diagonalization of Hermitian matrix

I would like to perform diagonalization of a Hermitian matrix $A$ and I know the steps but at the end I am not getting diagonal matrix with eigenvalues on the main diagonal, can anyone help me why? ...
0
votes
2answers
25 views

How to prove inverse of a linear operator is diagonalizable using concept of eigenspaces?

Let T be an invertible linear operator on a finite dimensional vector space V. Given for any eigenvalue $\alpha$ of T, $\alpha$^(-1) is an eigenvalue of T^(-1). I first proved that the eigenspace of ...
-2
votes
3answers
43 views

If T is diagonalizable then prove that T inverse is diagonalizable. [closed]

If T is an invertible linear operator on a finite dimensional vector space V, then if T is diagonalizable prove that T inverse is also diagonalizable.
0
votes
1answer
25 views

Eigenvectors for shear matrix and diagonalizing.

Here is a shear matrix $ \begin{pmatrix} 1 && 0 \\ 2 && 1 \end{pmatrix}$. The eigenvalues are 1. $ \lambda^2 - 2 \lambda + 1 \to \lambda = 1$. So now I try to find the eigenvectors. ...
5
votes
0answers
35 views

Diagonalization of a big scary matrix

I would need to diagonalize this tridiagonal block matrix $M$: $$M = \begin{bmatrix} A & B & & \\ B^T & A & B & \\ & B^T & A & B \\ & & \ddots & ...
0
votes
1answer
22 views

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix. $V = P_1(R)$ and $T(ax+b) = (-6a + 2b) x + (-6a + b)$ First i gave the canonical basis of ...
0
votes
0answers
31 views

matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
0
votes
0answers
18 views

Whitening of a 2x2 diagonalized matrix

I am having a a hard time finding a good example that will fully explain the final steps of taking a diagonalized matrix through the whitening process. I have the following values. A=(1, -3/4; -3/4, ...
-1
votes
0answers
37 views

Can't find a diagonal dominant matrix

I tried to solve system of linear equations using Jacobi method and one of the step is diagonally dominant matrix. My initial matrix is: $\begin{bmatrix}11.07 & 8.01 & -8.47 & 6.84\\16.65 ...
0
votes
4answers
53 views

Finding the diagonalizing matrix.

Find a nonsingular matrix $C$ such that $C^{-1}AC$ is a diagonal matrix. $$ A=\begin{pmatrix} 1 & 0 \\ 1 & 3 \\ \end{pmatrix} $$ I have found the eigenvalues to be 1 ...
1
vote
1answer
55 views

Finding the trace of $(I + \Sigma^{-1} AA^T)^{-1}$

I need to efficiently compute the trace of $$ B = (I + \Sigma^{-1} AA^T)^{-1} $$ where $\Sigma$ is diagonal and all its elements strictly greater than zero. $A$ is $-1$ on the diagonal and $1$ right ...
1
vote
2answers
59 views

Prove that S is diagonal

Let $S: V\rightarrow\ V$ be an operator on an $n$-dimensional real vector space with an eigenvalue that has geometric multiplicity equal to $n-1$. Prove that $S$ is diagonal. Give an example of such ...
1
vote
2answers
32 views

How to show CBC = I and CAC is a diagonal matrix for B positive definite and A positive semi definite?

How would you accomplish this: Show that if $A$ is a positive semi definite matrix and $B$ is a positive definite matrix, both $n\times n$, then there is a matrix $C$, also $n\times n$, such that ...
-1
votes
2answers
53 views

If a matrix is non diagonalizable, what other method can I use to calculate the nth power?

First off, I have this matrix A: 1 0 3 1 0 2 0 5 0 I have calculated the eigenvalues, which are ...
2
votes
3answers
75 views

Check diagonalizability of a matrix without using eigen properties

For the matrix $$ A=\begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} $$ How can we determine if $A$ is diagonalizable over $(a) \mathbb{F}^2 (b) \mathbb{Q} ...
1
vote
1answer
41 views

Matrix: determinant & Diagonal

There is a question that comes up in my mind after I watched Prof. Gilbert Strang's lectures. He was saying: For any matrix $A$, Since $A = LU$, $\det(A) = \det(LU)$ and $\det(L) = 1$, hence $\det(A) ...
0
votes
1answer
40 views

How do you diagonalize this matrix and find P and D such that A = PDP^-1?

1 1 4 0 -4 0 -5 -1 -8 I3 = 3x3 identity matrix λ 0 0 λI3 = 0 λ 0 0 0 λ λ-1 -1 -4 = 0 λ+4 0 5 1 ...
0
votes
0answers
25 views

Define a positive dot product in $\mathbb{R^3}$

Consider the matrix $A= \begin{bmatrix} k & k-1 & 0 \\ 1-k & 2-k & 0 \\ 2k-3 & 2k-1 & 2 \end{bmatrix} $ with $k \in \mathbb{R}$ and let be $f_a: \mathbb{R^3} \rightarrow ...
0
votes
2answers
54 views

$A^{T}A$ is diagonal. What can I say about $A$?

Is there any special property about the elements of $A$ if $A^{T}A$ is diagonal? I imagine you need some sort of symmetry but I can't see what it should be. Edit: Sorry, maybe it's better phrased ...
1
vote
2answers
63 views

Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...
1
vote
0answers
30 views

Rational canonical forms of a matrix

Let $A$ a $4\times 4$ matrix over the reals with $A^3-A^2+A=0$ That means that its minimal polynomial is: $x$ or $x^2-x+1$ or $x^3-x^2+x$ In the first case $A=0$. The second case is $$ ...
0
votes
0answers
44 views

Build up a not diagonalizable linear map

I need an hint for this problem. Let be $M = \begin{bmatrix}2 & 1 \\ -2 & 0\end{bmatrix} \in M_2(\mathbb{K})$ and $H=\{A \in M_2(\mathbb{K}) : AM=MA \} $ Build up a linear map $f: ...
0
votes
3answers
53 views

Finding the eigenvectors and the diagonal of a singular 2x2 matrix

i am trying to find the eigenvectors of a 2x2 singular matrix, A = [0 , 1 ; 0 , -3]. My problem is that i can't. I know the answer is, Q = [1 , 1 ; 0 , -3] (by using Matlab), but i don't understand ...
1
vote
2answers
59 views

Matrix diagonalization and operators

Let $V=\Bbb F^{m\times n}$ $T: V\to V$ , by $T(B)=P^{-1}BP$ , for any $B$ in $V$ , where $P$ is an invertible matrix. prove that if $A$ is an eigenvector of $T$, with eigenvalue $\lambda$ and $A$ is a ...
0
votes
1answer
41 views

Diagonalization of a matrix with change of basis

I was trying to diagonalize a not really nice matrix doing first a change of basis but I noticed that the two characteristic polynomials I get are different. Original matrix and its characteristic ...
0
votes
0answers
21 views

Diagonalization of sparse block matrix

I have a real symmetric matrix, \begin{equation} \left( \begin{array}{ccc} 0 & M & M' \\ M ^T & 0 & 0 \\ M ^{ \prime T} & 0 & 0 \end{array} \right) \end{equation} ...
2
votes
1answer
45 views

Complex matrix and diagonalizablity

Let $A\in\mathcal{M}_4(\mathbb C)$ such that $\operatorname{rank}(A)=2$ and $A^{3}=A^2$ $\neq0$. Suppose that $A$ is not diagonalizable. Then 1. One of the Jordan blocks of the Jordan cannonical form ...
2
votes
3answers
29 views

Rank of a diagonalizable matrix?

What can be said about the rank of a diagonalizable matrix?
0
votes
2answers
38 views

Diagonalizing a matrix. Which formulae is correct?

In my coursebook on linear algebra on some page I see that a diagonal matrix $D$ for a matrix $A$ that can be diagonalized ca be found as follows: $$\tag{1}D=T^TAT$$ But reading further I see that my ...
0
votes
1answer
45 views

Trouble understanding the diagonal matrix theorem.

The Diagonal Matrix Representation Theorem states: Suppose $A=PDP^{-1}$, where $D$ is a diagonal $nxn$ matrix. If $B$ is the basis for $R^n$ formed from the columns of $P$, then $D$ is the $B$-matrix ...
3
votes
3answers
73 views

Prove that $T^n$ is diagonalizable.

Prove or give a counterexample: If $V$ is a complex vector space and $\text{dim V} = n$ and $T \in L(V)$, then $T^n$ is diagonalizable. In order to show that $T$ is diagonalizable I need to show ...
1
vote
3answers
40 views

non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?
0
votes
1answer
37 views

Eigen vectors of the matrix whose columns are eigen vectors of the original matrix

Consider a matrix $A$ of dimension $n$X$n$ whose eigen vectors are $y_1,y_2,y_3,...,y_n$ and are linearly independent. What are the properties of the eigen vectors of the matrix $P$ whose columns are ...
2
votes
1answer
86 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
3
votes
1answer
64 views

If diagonalizable matrices commute does it neccesarily mean that they can be simultaneously diagonalized?

If matrices $M_1$ and $M_2$ can be simultaneously diagonalized, than they commute, which can be easily shown: \begin{align} M_1M_2&=P^{-1}D_1PP^{-1}D_2P \\ &=P^{-1}D_1D_2P \\ ...
1
vote
3answers
54 views

Diagonalizability of a certain $4\times4$ matrix

Question $\bf 3.$ Determine if the following matrix is diagonalizable. (explain your answer) $$A=\pmatrix{ 1 & 4 & -2 & 3 \\ 3 & -3 & 0 & 4 \\ 1 & 1 & 1 ...
1
vote
0answers
38 views

Show that isotropic function S(A) and A have same eigenvectors

Given $\boldsymbol{A}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{S}(\boldsymbol{A})\boldsymbol{Q}^T = \boldsymbol{S}(\boldsymbol{QAQ}^T)$ $\forall ...
1
vote
3answers
49 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
2
votes
2answers
43 views

diagonalization of a matrix over finite fields

I'm having a problem with determine whether a matrix is diagonalizable over $\mathbb F_{2}$, over $\mathbb F_{3}$, etc. for example, for the following matrix: $$ \begin{bmatrix} 1 ...
3
votes
0answers
87 views

Simultaneous diagonalization of commuting matrix

I have 3 diagonalizable matrices $A,B,C$. They commute with each other $[A,B]=[B,C]=[A,C]=0$ [edit] The matrix $A$ is Hermitian but $B$ and $C$ have no properties. [/edit] I can get the eigenvalues ...
1
vote
0answers
20 views

Fast way to find exponential of a matrix dot product where one of them is diagonal

Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ ...
0
votes
1answer
28 views

orthogonal matrix

I have to show the following claim: Let $A\in Mat(n,\mathbb{R})$ be positive definite and symmetric. Show that there exists a Matrix $T\in Mat(n,\mathbb{R})$ such that $T^tAT$ is a diagonal matrix. My ...
1
vote
2answers
111 views

$A^2$ is diagonalizable leads to $A$ diagonalizable?

If $A^2$ is diagonalizable, is it necessary true that $A$ is diagonalizable? Also, the opposite: If $A$ is diagonalizable, is it necessary true that $A^2$ is diagonalizable? I'm not sure yet, tried ...
0
votes
3answers
90 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...