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.
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Diagonalization problem, linear algebra
Probe that if $A$ is an invertible matrix, then $\frac{\chi_A(\lambda)}{\det(A)} = (-\lambda)^n \chi_{A^{-1}} (\lambda^{-1})$, where $\chi_A(\lambda)$ is the characteristic polynomial of the matrix $A$...
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Eigenvectors, Eigenvalues, Diagonalization
1. Let $T : R^n \to R^n$ denote a linear transformation, and let
$\vec{v} \in R^n$ denote a non-zero vector satisfying $T(\vec{v}) = \lambda \vec{v}$ for some number $\lambda$.
a) Verify that $T^n(\...
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How to compute the eigenpairs of a real symmetric matrix extended with one row/column pair in the most effective way?
I have a real $n \times n$ symmetric matrix $\mathbf{A}_{n \times n}$ which I would like to extend with a row/column pair:
where $\alpha$ and $\beta$ are two real numbers and $\mathbf{0}_n$ is a zero ...
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Are all eigenvectors of a symmetric matrix $\breve{K}$ given by the eigenvectors of $\breve{K}^2$?
I aim to compute the eigenvectors of a symmetric matrix $\breve{K}$ (here it is actually $\in \mathbb{R}^{4 \times 4}$)
$$\breve{K} \vec{u} = \lambda \vec{u}.$$
I found that the matrix $\breve{K}^2$ ...
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Diagonalization and Homomorphism
Maybe this question could sound silly, but after carefully re-reading my linear algebra notes one particular detail catch my sight.
Let $V$ a $\Bbb K$-vector space, $\dim(V) = n$, $f$ an $\textbf{...
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Eigenvalues, Eigenvectors and Diagonalisation of Matrix A
I have the following question, from a Linear Algebra exam paper:
I have found the eigenvalues and vectors as well as the basis:
$ \lambda_1=2,\lambda_2=2,\lambda_3=1\\
\\
v_1=(-1,0,1)\\
v_2=(0,1,0)\\
...
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If $ A $ and $ A^T $ commute, do they share the same eigenvectors when they are not diagonalizable?
I saw this on Strong's book Introduction to Linear Algebra, but it didn't prove it:
If $ A $ and $ A^T $ commute, then they share the same eigenvectors.
And now I am wondering whether it is still ...
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If minimal polynomial of a matrix A is irreducible, is A diagonalizable?
Given A is a $7\times7$ matrix with characteristic polynomial $ (\lambda -3)^5 (\lambda-1)^2$ and if minimal polynomial of A is irreducible, is A diagonalizable?
My attempt: I am not sure of what ...
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Jacobi rotation for large matrices is not giving answers everytime [closed]
I am from chemistry background. I tried to write a code to diagonalize matrices by Jacobi rotation. Here is my code,
...
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Conditions for a complex matrix to be diagonalizable
I have a family of $100 \times 100$ complex symmetric matrices. I need to check whether they are diagonalizable and the calculation of eigenspaces is computationally expensive.
I was told about the ...
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What does it mean for a linear operator to have eigenvectors but not be diagonalizable? [closed]
Could someone explain this conceptually? For example, giving a simple example with $\mathbb{R}^3$
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Diagonalize the matrix $A$ or explain why it can't be diagonalized where $\lambda=1$
The matrix $A$ is $$\begin{bmatrix}1&0 \\ 6 & -1 \end{bmatrix}$$
For a matrix to be diagonalizable it can be rewritten as $$A=PDP^{-1}$$
where $P$'s columns are the eigenvectors of $A$ and $D$ ...
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Can we always off diagonalize matrices with purely imaginary eigenvalues?
Consider a two-by-two matrix $A$ which has real entries with purely imaginary eigenvalues. I am wondering if one can deduce that $A$ under some basis is given by
$$
\begin{pmatrix}0 & \lambda \\ -\...
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What is the fundamental property of the Fourier operator?
It is well-known that the Fourier transform satisfies the following translation property:
$$[\mathcal{F}(\mathcal{T}_hf)](\xi) = \text{e}^{2 \pi \text{i} \xi h} \cdot [\mathcal{F}(f)](\xi),$$
where $\...
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How to use the minimal polynomial theorem for this block matrix
Let $M$ be a matrix made up of two diagonal blocks: $M = \begin{pmatrix} A & 0 \\ 0 & D\\ \end{pmatrix}$
Prove that $M$ is diagonalizable if and only if A and D are diagonalizable.
I know I'd ...
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Prove this theorem from linear algebra. [closed]
Theorem: If $\{e_1,\ldots,e_n\}$ is an orthonormal basis of $V$ such that each $e_j$ is an eigenvector, then the matrix of $A$ with respect to this basis is diagonal,
and the diagonal elements are ...
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Wave equation, PDE Course
I'm taking a course in PDE's, and I'm trying to understand an example from the course literature, and would like some help.
The example is the following
Example 1: (baby wave equation)
Consider the ...
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Diagonalization of a block matrix with (almost) Toeplitz blocks
Background
Consider the matrix $R \in \mathbb{R}^{12 \times 6}$ whose structure is given below as:
This represents a discrete gradient operator for a $2 \times 3$ grid equipped with reflexive ...
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If $A=M^{-1}A^tM$ for some $M>0$, then there exists an invertible matrix $P$ such that $P^{t}MP=I$ and $P^{-1}AP$ is diagonal.
Problem
Let $V=\mathbb R^n$ be the space of column vectors, and $M$ a positive definite symmetric $n\times n$ real matrix. Suppose the matrix $A\in M_n(\mathbb R)$ satisfies $MAM^{-1}=A^t$. Show that ...
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Difference Between Simultaneous Diagonalization and Generalized Eigenvalue Problem
I am wondering about the difference between simultaneous diagonalization and the general eigenvalue problem. Here is my understanding of simultaneous diagonalization and the general eigenvalue problem:...
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What is the state of the art algorithm for diagonalizing real symmetric matrices?
There are many methods for diagonalizing matrices; probably the most widely used is the combination of household transformations and the QR algorithm.
Is there any superior method for diagonalizing ...
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Diagonalization matrix and relate to abstract algebra
A matrix $A\in M_n(K)$ is diagonalizable iff $\exists\hspace{.03cm}P\in GL_n(K):\ P^{-1}AP$ is diagonal matrix. Then $B=P^{-1}AP$ is called similar to $A$. I realize this is the conjugate element ...
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If $f:\ V\longrightarrow V$ be linear then how the fact $V=ker\,f+im\,f$ relate to the diagonalization / Jordanization of $f$?
Let $V$ be a n-dimensions vector space and $f$ be a linear map on $V$. Then is there any relevant between $V=ker\,f+im\,f$ and the diagonalization/Jordanization of $f$ ?
Thanks
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If $A$ is diagonalizable and $B$ commutes with all transformations commuting with $A$, does it follow that $B=p(A)$? [duplicate]
Let $A$ be a diagonalizable transformation on a finite-dimensional vector space. If $B$ commutes with all transformations commuting with $A$, does it follow that $B=p(A)$ for some polynomial $p$?
...
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If $A,B$ are invertible commuting $2\times 2$ matrices, and $AB$ is diagonal but not multiple of the identity. Are A,B both diagonal?
The problem is as in the title. I think the answer is yes.
There is a similar question here: If $AB=BA$ and $AB$ is diagonal are $A$ and $B$ both diagonal?
But the counterexamples are either non-...
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Relation between eigenvalues and rank of matrix
Prove/Disprove: Let $A$ be a square matrix of order $n $ then rank of $A$ is atleast number of non zero eigenvalues of $A.$
My approach: For any matrix $A$, $A^TA$, and $AA^T$ are both symmetric and ...
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Block matrices. Each submatrix is a permuation of an original matrix at the diagonal.
Suppose that we have a matrix M which consists of block matrices with the same dimensions.
\begin{equation}
M=\begin{pmatrix}
A&D&0&0\cdots&0&0\\
D&P^{-1}AP&D&0\...
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Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathcal{R}^N$ to $\mathcal{R}^N$. For each $x\in \mathcal{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as
$$
J_f(x)=S(x)\Lambda(x) {S(x)}^{-1},
$$
where the ...
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Discussion about diagonalizing matrix
I have equation
$I = \rho(r)^{T} A_{h} \rho(r)$,
where
$\rho(r)= \begin{pmatrix}\rho_{+}(r) \\ \rho_{-}(r) \\ \rho_{z}(r)\end{pmatrix}$.
I want to rearrange $I$ by diagonalizing $A_{h}$ such as
$\...
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SVD of product of diagonal and unitary matrices
Given two (possibly rectangular) diagonal matrices $\Sigma_\text{L}$ and $\Sigma_\text{R}$ with nonnegative elements, what can we say about the singular value decomposition of
$$\Sigma_\text{L} X \...
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Determining values of a parameter for which a matrix is complexly diagonalizable
Given is the $3 \times 3$ matrix $A$ below. I want to know for which values of $c$ the matrix is complex diagonalizable. Based on my understanding, a matrix is complex diagonalizable if and only if ...
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Help me to improve this proof of the spectral theorem of Hermitian matrices
Spectral theorem for Hermitian matrices states that, for an $n\times n$ Hermitian matrix $A$:
a) all eigenvalues are real, b) eigenvectors corresponding to distinct eigenvalues are orthogonal, c) ...
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Concatenated diagonalization of combined real/imaginary diagonalization
I have a problem involving a complex matrix that I need to diagonalize and apply weights to the entries according to a function relating only to the eigenvalues of the real part of the matrix.
Suppose ...
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A question about the criterion of diagonalizable normal operator on a Hilbert space (perhaps non-separable).
This is the excercise 2.11 in Chapter IX of A Course in Functional Analysis by John B. Conway:
Suppose $H$ is a Hilbert space,$\,N$ is a bounded normal operator with associated spectral measure $E.$ ...
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Values of a,b,c that make a matrix diagonalizable
For which, if any, values of $a,b,c \in \mathbb{R}$ is the matrix
diagonalizable?
The Eigenvalues are easy enough. I get $\lambda=1$ (With an algebraic multiplicity of $2$) and $\lambda=c$ (With an ...
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Need Help Understanding and Solving Matrix Factorization Problems
I hope you're doing well. I'm currently working on a set of matrix factorization problems and could really use some assistance in understanding and solving them. I've been trying to make sense of the ...
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Intriguing Tridiagonal Matrix
Given a positive sequence $\{a_n\}_{n=1}^N$, please consider the matrix
$$ \begin{bmatrix}
a_1 & -a_1 & & & & \\
-a_1 & a_2 + a_1 & -a_2 & & & \\
& - ...
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Can we continuously unitarily diagonalize a symmetric matrix?
Let $S(t)$ be a mapping on $[0,1]$ into the space of symmetric real matrix with usual topology, if $S$ is continuous on $[0,1]$,
then can we show the existence of continuous mappings $Q(t), \Lambda(t)$...
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Counting diagonalizable matrices?
Consider the set $$A=\{{0,1,2,3,...,27}\}$$ I am required to find the number of matrices \begin{bmatrix}
a & b\\
0 & c
\end{bmatrix} where $a,b,c\in A$ and the matrix is diagonalizable.
Here ...
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ordering of eigenvalues for simultaneously diagonalizable matrices [closed]
Suppose I have two commuting Hermitian matrices $A$ and $B$: $[A,B] = 0$. I can always find a unitary operator $U$ such that simultaneously diagonalize both matrices, i.e.,
\begin{equation}
U^* A U = ...
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LADW — The condition that ensures a real matrix $A$ to have real diagonalization [closed]
In chapter 4 of Sergei Treil's Linear Algebra Done Wrong (LADW), the author first introduce the theorem that an operator $A: V \to V$ is diagonalizable iff for each eigenvalue $\lambda $, the ...
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If $A$ is positive real symmetric matrix and $B$ is real symmetric, does there exist some $V$ such that $V^{T}AV$=$I$ and $V^TBV$ is diagonal matrix?
Let's say we have two matrices $A$ (a positive real symmetric matrix) and $B$ (a real symmetric matrix). And let us suppose that in general $A$ and $B$ don't commute with each other. Then,
Q. Is it ...
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Can we prove $T : R^n \rightarrow R^n$ $T(x_1,x_2,...x_n) = ((x_1+x_2+...x_n),...,(x_1+x_2+...x_n))$ ,is diagonalizable by induction (by finding e.v)?
Let $T : {\Bbb R}^n \to {\Bbb R}^n$ be defined by
$$ T(x_1, x_2, \dots, x_n) = ((x_1 + x_2 + \cdots + x_n), \dots,(x_1 + x_2 + \cdots + x_n))$$
It is possible to prove that $T$ is diagonalizable by ...
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A nicer proof of the diagonalisation criterion via module theory?
I'm wondering if we can apply modules structure theory to prove a standard statement about vector spaces over a field $F$: an endomorphism $\alpha : V \rightarrow V$ is diagonalisable $\iff \exists$ a ...
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Finding an orthogonal and diagonal matrix given eigenvalues/vectors
So for a symmetric matrix A I am given the eigenvalues $\lambda_1=3$ and $\lambda_2=-1$ where {[1 2 -1],[0 1 1]} is a basis for $E_3$ and {[-3 1 -1]} is a basis for $E_{-1}$.
I have the diagonal ...
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Operators on Finite Complex Vector Spaces
Let $S$ and $T$ be linear operators on a finite-dimensional vector space $V$ over $\mathbb{C}$. Prove or disprove the following:
If $ST$ is diagonalizable, then both $S$ and $T$ are diagonalizable.
...
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Given Ker(T) and Ker(T-2I), is T diagonalizable?
I'm stuck on this problem that I found on my book of linear algebra.
Be $T:\mathbb{R}ˆ3 \rightarrow \mathbb{R}ˆ3$ a linear map such that
$$Ker(T-2I) = \{(x,y,z) | x+y=0\} $$ and $$Ker(T) = \{ (x,y,z)|...
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How to diagonalize a matrix with a non-numeric element?
For what $\alpha$ is the matrix A diagonalizable, where
$$ A =\begin{pmatrix}
1 & 0 & 0 & \\
\alpha-3 & 6-2\alpha & 6-3\alpha \\
2-\alpha & 2\alpha -4 & 3\alpha-4 \\
\...
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If A is row vector 1×n then is $A^tA$ always diagonizable?
If $A$ is row vector 1×n then is $$A^tA$$ always diagonizable?
Multiplication of $A$ transpose times $A$ gives a matrix of $n×n$ and i was able to prove that this matrix is diagonizable over $\mathbb ...
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Find a non-square matrix satisfying $CA=BC$
Given two matrices $A\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{4\times 4}$, which don't have the same eigenvalues. Suppose there exists a matrix $C\in\mathbb{R}^{4\times 3}$ satisfying $CA=BC$. ...
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