Questions tagged [diagonalization]

For questions about matrix diagonalization. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. This tag is NOT for diagonalization arguments common to logic and set theory.

Filter by
Sorted by
Tagged with
0
votes
3answers
40 views

For which values is the matrix diagonalizable?

i have the following 2x2 matrix: $\left( \begin{array}{rrr} 0 & 1 \\ -1 & a \\ \end{array}\right)$, with $a ∈ \mathbb{R}$ and then same matrix but with $a ∈ \mathbb{C}$ For which $a$ is the ...
0
votes
1answer
12 views

Is $((w_i, Hv_j))_{ij}$ diagonalizable?

Let $(V, (\cdot,\cdot))$ be a finite-dimensional inner product space, and let $H$ be a Hermitian operator on $V$. We know that $H$ is diagonalizable, and its eigenvalues are real. We consider the ...
4
votes
1answer
94 views

If diagonalizable matrices are not dense over $\Bbb R$, how common are they?

The link: The diagonalizable matrices are not dense in the square real matrices says that diagonalizable matrices are dense over $\Bbb C$ but not over $\Bbb R$. If that's the case, then the logical ...
2
votes
2answers
69 views

Shouldn't the matrix of eigen vectors of a defective matrix be non-invertible?

Is it true that a defective (non-diagonalizable) square matrix has a set of eigen vectors that don't span the whole space? In that case, if we do its singular value decomposition, shouldn't the matrix ...
2
votes
1answer
120 views

Proving a matrix is diagonalizable given eigenvectors and information about characteristic polynomial ranks

Let $A \in \Bbb R^{5 \times 5}$. Let $$v_1=(1,0,0,1,1), \quad v_2=(1,1,0,0,1), \quad v_3=(-1,0,1,0,0)$$ be eigenvectors of $A$. Also, $$\rho(2I-A) \gt\rho(3I-A)$$ and $$A(1,2,2,1,3)^t=(0,4,6,2,6)^t$$ ...
0
votes
2answers
33 views

Diagonalized matrix not zero on sidelines

I try to diagonalize: $Y = \left(\begin{array}{ccc} X & -X & 0\\ -X & X & 0\\ 0 & 0 & 2\,X \end{array}\right)$ after doing the general diagonalization formula: $Y_{\mathrm{...
1
vote
0answers
39 views

Given matrix $A$ , need to find an invertible matrix $P$ and a diagonal matrix $D$ such that $D=P^{-1} \cdot A\cdot P$

Given $A= $$\left(\begin{matrix} 2 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \\ \end{matrix} \right) $ find an invertible matrix $P$ and a diagonal matrix $D$ such that $D=P^{-1} \cdot ...
0
votes
0answers
20 views

Diagonalize a generic non-symmetric complex matrix via sandwitching with two unitary matrices.

Suppose $M$ is a generic rank-$N$ complex-valued non-symmetric matrix, so its entry $M_{ij} \in \mathbb{C}$. How to prove mathematically rigorously that it is always possible to (1) find unitary ...
1
vote
1answer
66 views

Diagonalisation of stochastic matrices

Suppose that $(X_n)_{n≥0}$ is a Markov chain on a state space $I = {1, 2}$ and stochastic matrix $$P = \begin{bmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{1}{3} & \frac{2}{3} \end{bmatrix}$$ (a)...
0
votes
1answer
28 views

how to solve this third degree characteristic polynomial?

(this exercise is like the previous question I've written today, but now I have a 3-by-3 matrix (with a real parameter $k$). I need to say where the matrix could be diagonalized, if it's possible. (...
0
votes
0answers
28 views

I'm having a problem when trying to diagonalize a matrix (with a parameter), how can I solve?

I have this matrix, and I need to say where the matrix could be diagonalized. Alpha is a real parameter. $$ \begin{matrix} 1 & \alpha \\ 3 & 1 \\ \end{matrix} $$ I found characteristic ...
1
vote
0answers
22 views

Prove $n\times n$ matrix with uniform random elements is diagonizable?

Let $X$ be a $n\times n$ matrix with $n\in\mathbb{N}^{+}$ and all off diagonal elements of $X$ following a uniform distribution $x_{ij}\sim\mathcal{U}(0,1)$ for any $i,j\in\{1,...,n\}$ with $i\neq j$. ...
0
votes
2answers
20 views

Unitarity similarity transformation

Prove that $R$ and $R^{\dagger}$ can be diagonalized by a common unitary similarity transformation if $R^{\dagger}$ is commutable with $R$. Let $R = SMS^{-1}$, where $M$ is diagonal and $S$ is ...
3
votes
0answers
178 views
+50

Help verifying proof that symmetric matrices are diagonalizable.

We have the standard proof that a symmetric matrix can be diagonalized even if it has eigen values with algebraic multiplicity via induction here. My boss came up with an alternate proof of this that ...
0
votes
1answer
44 views

Does the matrix $U$ that diagonalize another matrix $M$ also diagonalize the derivative of $M$?

I have a $3\times3$ Hermitian matrix $M$. All the elements of $M$ are a function of variable $x$. There is a unitary matrix $U$ that diagonalize $M$, i.e. $$ D=U^\dagger MU $$ I wonder if the same $U$ ...
2
votes
3answers
45 views

What are the possible values of the sum of the dimensions of the eigenspaces of an $n \times n$ matrix over $\mathbb{R}$?

My question has two parts: First, given an $n$th degree polynomial $p$ over $\mathbb{R}$, is there necessarily an $n\times n$ matrix whose characteristic polynomial is $p$? If so, then would it be ...
5
votes
0answers
56 views

Diagonalizing a $4\times4$ traceless Hermitian matrix with Lie groups

Before I talk about $4\times 4$ matrices, let me first provide some background using the example of $2\times 2$ traceless Hermitian matrices $H$. I won't write in full rigor since I think the ...
0
votes
0answers
10 views

Error incurred upon truncating a diagonal matrix to only the k largest eigenvalues.

Say I have a diagonal matrix (A) of dim n x n. What is the error incurred if I approximate it with only its k-largest eigenvalues? I am using the 1-norm. I am trying to quantify the error in the ...
4
votes
1answer
94 views

Changing the basis in diagonalization: why doesn't it work?

I read the theorem on Apostol Calculus II : Theorem $4.10$ Let $T : V \to V$ be a linear transformation, where $V$ has scalars in $F$, and $\dim V = n$. Assume that the characteristic polynomial of $...
1
vote
2answers
52 views

$C=A+B$ diagonalizable $\implies$ $A$ and $B$ simultaneously diagonalizable?

If $A$ and $B$ are simultaneously diagonalizable matrices, then clearly $A+B$ is diagonalizable. But is the converse true ? i.e if $C=A+B$ is diagonalizable, are $A$ and $B$ simultaneously ...
1
vote
1answer
24 views

$A,B$ be $n\times n$ semipositive-definite, rank $A=r.$ Find an invertible $P$, such $P^{-1}A(P^{-1})^T=diag(I_r,0)$, $P^T BP=diag(\lambda_i)$

$A,B$ be $n\times n$ semipositive-definite, $\operatorname{rank} A=r$. Find an invertible $P$, such $P^{-1}A(P^{-1})^T=\operatorname{diag}(I_r,0)$, $P^T BP=\operatorname{diag}(\lambda_i)$ Here, $I_r$ ...
0
votes
1answer
48 views

Set of diagonalizing matrices

Let $M \in \mathbb{R}^{n \times n}$ be a positive definite matrix. I would like to characterize the set $$ \mathcal{A} := \{A \in \mathbb{R}^{n\times n} : AMA' \text{ is diagonal and invertible} \}. $$...
-1
votes
0answers
42 views

Proof that for a diagonalizable matrix $A$, $A^n=PD^nP^{-1}$ for $n\notin\mathbb{N}$?

In the past, I’ve seen proofs that assert that if a square matrix $A$ is diagonalizable and can be written in the form $A=PDP^{-1}$ where $D$ is a diagonal matrix whose diagonal entries contain the ...
0
votes
0answers
42 views

If $A$ is a real symmetric matrix with full rank and only simple roots, what can be said about $|B\otimes I+I\otimes B|$, for $B = \sum_k a_k A^k$?

I'm considering a situation in which $A, B > 0$ are real $(p\times p)$ symmetric matrices where $A$ has $p$ distinct eigenvalues $\{\lambda_i\}_{i=1}^p$, and $AB=BA$. I'm interested in the level of ...
1
vote
1answer
31 views

diagonalising (I-X(X^TX)^-1X^T where X is rank n-k, a square matrix dimension nxn, into a matrix with n-k 1 along the diagonal, and then zeros.

For context, this query relates to the derivation of the distribution of the variance estimator in a linear regression. I’m doing this in a 3rd-year econometrics unit, and which takes a very matrix ...
0
votes
0answers
38 views

Continuity of a basis of a family of quadratic forms

Let $p$ be a parameter in a k-sphere $\mathbb{S}^k$ and let $Q^p$ be a family of real quadratic forms over $\mathbb{R}^n$, which is continuous with respect to $p$. For every fixed $p$, there exists an ...
0
votes
0answers
28 views

Absorbing generator matrix in continuous-time Markov chain models

Setting Let $[0,T]$ with $T\in\mathbb{R}^{+}$ be a time horizon over which $N\in\mathbb{N}^{+}$ continuous-time time-homogeneous Markov chains make transitions between $\{1,...,h\}$ states with $h\in\...
2
votes
1answer
99 views

Does $A^T A$ have complex eigenvalues?

I learned about simple value decompositions from my online Linear Algebra class today, and one thing I learned in particular was that for any real $m \times n$ matrix $A$, $A^TA$ has only nonnegative ...
0
votes
0answers
55 views

Find P with $PAP^{-1}=D$ with no calculations

I have this exersize and we were told these 2 matrices can be diagonalized with no calculations at all. We need to find P such that $PAP^{-1}=D$ is diagonal (if the matrices are diagonalizable) for ...
1
vote
0answers
34 views

Eigenvalues/vectors of combinations of Gell-Mann matrices

Let's consider the Gell-Mann matrices $\vec{\lambda} =(\lambda_1, \lambda,_2,\cdots, \lambda_8)$. Another hermitian matrix can be constructed in terms of a real vector $\vec{a}=(a_1,\cdots,a_8)$ by $A ...
0
votes
1answer
25 views

Computing a closed formula for a recurrent sequence using eigen -values and -vectors

How would you use eigenvalues and eigenvectors to compute a closed formula for the following sequence: $$\{x_0=1, x_1=2, x_n=5x_{n-1} + 14x_{n-2}, n \ge 0 \}$$ I have come up with the following ...
1
vote
0answers
23 views

How is this equivalent: $[Tv_i]_B = λ_i[v_i]_B = λ_ie_i\,$?

The question is: There is a basis of $V$ consisting entirely of eigenvectors of $T$ $\implies$ T is diagonalizable, where $T$ be a linear operator over $V$ with a basis $B$. I've looked at the ...
0
votes
1answer
47 views

Why this matrix is not diagonalizable? [closed]

Let be $A$ a $n\times n$ matrix such that, rank($A)=n-1$ and rank($A^2)=n-2$. ¿Why a matrix like that is not diagonalizable?
2
votes
1answer
47 views

Relation between eigenvalues and matrix's characteristic

If I let $A$ be a $4×4$ real matrix with eigenvalues $−1, 1, 2, 3$ I am having hard time figuring out matrix characteristic! Is matrix A invertible? (I think it is invertible since it has no $0$ ...
0
votes
2answers
58 views

For a linear map $f: V \to V$ if $f^2$ is diagonalizable and $\ker f = \ker f^2$ then is $f$ diagonalizable?

Here $V$ is a finite dimensional vector space, of dimension $n$, over an algebraically closed field $F$. My original approach was to use a minimal polynomial argument by showing that $\pi_f$ (which I ...
0
votes
1answer
34 views

Hermitian Matrix eigenvalues

I am trying to show the following sentence: Let H be hermitian with $\sigma(H) \subseteq \lbrace-r,r\rbrace$. Show that H ∘ H = $r^2$* I. Since H is hermitian, we know that we can decompose the matrix ...
1
vote
0answers
36 views

Diagonalization, algebraic and geometric multiplicity

Let $T: \Bbb{R}^4 \rightarrow \Bbb{R}^4 $ given by $T(x,y,z,w) = (x+y,y+z+w,z+w,z+w)$. Determine whether T is diagonalizable. If so, determine a basis $\beta$ $\Bbb{R}^3$ formed by eigenvectors of T ...
1
vote
1answer
32 views

When is a complex symmetric matrix with only the last row and column being non zero diagonalizable?

Imagine the $n \times n$ matrix A: $$\begin{bmatrix} 0 & 0 &... & 0 & a_1\\ 0 & 0 & ...& 0 & a_2\\ . & .&&.&.\\ a_1 & a_2 & ... & 0 &a_n ...
1
vote
1answer
26 views

Change of basis matrix exactly one row away from being correct

For homework, I'm given a matrix $$ A = \begin{bmatrix} 3 & 2i & -2i\\ -2i & 0 & -1\\ 2i & -1 & 0 \end{bmatrix} $$ in an hermitian space. We are trying to find an orthonormal ...
0
votes
1answer
47 views

One matrix is diagnolized by orthonormal basis of another matrix

Let $A\in \mathbb R^{n\times n}$. Suppose $B$ is symmetric and positive definite, and \begin{equation}\label{eq:sym} A^TB=BA, \end{equation} then $A$ is diagnolizable by a B-orthonormal basis. ...
0
votes
1answer
31 views

Question about inverse of a matrix.

Consider the block upper triangular matrix $$A = \left[ \begin{matrix} A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} \right], $$ where $A\in\mathbb R^{n\times n}$ and $A_{11}\in\mathbb R^{k\times ...
0
votes
1answer
35 views

Diagonalising a matrix: does order of eigenvectors matter?

So suppose I have a matrix $A$ with real distinct eigenvalues, and I am finding the diagonal matrix $D$ such that $D = P^{-1}A P$. Then $P$ consists of columns that are eigenvectors of $A$. In what ...
0
votes
0answers
24 views

Every real, skew-symmetric matrix is diagonalisable by a unitary matrix

I need to show, that every real, skew-symmetric matrix M can be diagonalized by a unitary matrix U. $$ M=-M^T \implies M = U D U^\dagger \quad \textrm{with} \quad U U^\dagger = U^\dagger U = \mathbb{I}...
1
vote
0answers
28 views

Orthogonal matrix and trigonometric functions of matrices

Let me try to summarize my question by taking into consideration some elementary notions of orthogonal matrices. The general $2\times2$ orthogonal matrix has the form: $$O = \begin{pmatrix} \cos\theta ...
1
vote
1answer
24 views

compute geometric multiplicity when $A$ is over field $\mathbb F_p$

Define $A=\begin{pmatrix} 0 & 0 & 3\\ 1 & 0 & 1\\ 0 & 1 & -3 \end{pmatrix}$ over $\mathbb F_p$.I am asking to find whether $A$ is diagonalizable when $p=2$. I have find out ...
6
votes
2answers
111 views

Orthogonal eigendecomposition of self-adjoint operator with indefinite scalar product

Let $V$ be a real vector space, of finite dimension $d$, equipped with a nondegenerate symmetric bilinear form $q$, and let $A$ be a linear map $V\to V$ that is self-adjoint with respect to $q$, i.e., ...
0
votes
0answers
23 views

Diagonalization and dimension of the eigenspaces of a matrix $A$

If a matrix $A$ is diagonalizable, what does this imply involving the dimensions of the eigenspaces of $A$?
4
votes
0answers
56 views

Fourier series of iterated sin / Diagonalization of an infinite matrix of Bessel functions

We define the iterated sine function as : $$ \sin^n(x) = \sin(\sin(.... \sin(x)))\:\:n\:\text{times.} $$ We know the "Frequency Modulation" formula based on Bessel functions :$$ \sin( p\, \...
1
vote
0answers
37 views

Orthogonal Diagonalization and Non-Symmetric Matrices

Given an $n \times n$ matrix $A$ is diagonalizable where, $A$ is non-symmetric i.e. $A^T\not= A$ And let $S = \{ \vec{v}_1 \,,\, \ldots \,,\, \vec{v}_n\}$ the set of the eigenvectors of $A$ , ($S$ ...
2
votes
0answers
28 views

Diagonalizing rank 1 correction of a diagonal matrix

Let $D$ be a diagonal matrix and $A$ be a rank 1 matrix. I want to compute the matrix exponential $ e^{D+A}$, if possible by diagonalization of the matrix $D+A$. Is there a way of computing the ...

1
2 3 4 5
44