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Question 1

4votes

2answers

237views

Why is a real symmetric matrix diagonalizable?

Exactly, I can't understand the real symmetric matrix is diagonalizable only from the symmetry.I can prove that the diagonalization of this kind of matrices by mathematical induction,as in Artin's ...

Rei Rockey

53

askedNov 12 at 14:01

Question 2

1vote

2answers

140views

Can one construct a liar's sentence using the diagonal lemma and the principle of explosion?

Isn't it possible to form the liar's sentence by using the diagonal lemma to form the sentence "This sentence proves all statements"?I.e. For some classical first order theory $T$ ...

William Oliver

1,262

askedNov 6 at 3:19

Question 3

3votes

1answer

237views

Inductive proof of necessary and sufficient conditions for diagonalisability

I am very confused by the following highlighted lines in a proof of necessary and sufficient conditions for diagonalisability in Linear Algebra Done Right (4th ed.), Axler S. (2024).Questions.How ...

microhaus

1,106

askedNov 2 at 1:19

Question 4

0votes

0answers

35views

Definite unimodular form equivalent to a diagonal form while itself being non-diagonalizable

In a lecture on exotic $\mathbb R^4$s, Robert Gompf claims that the following bilinear forms are "equivalent" (I presume, over the integers):$$\left(\begin{array}{c|c}1 & 0\\\hline0 &...

ruza-net

365

askedNov 1 at 21:59

Question 5

2votes

2answers

228views

Showing that two $2\times 2$ matrices are similar

I want to prove that the following $2\times 2$ matrices are similar over $\mathbb R$:$$A=\begin{pmatrix}0 & -1\\1 & 0\\\end{pmatrix}\qquadB=\begin{pmatrix}-3 & -5\\2 & 3\\\...

boaz

5,715

askedOct 2 at 20:57

Question 6

2votes

1answer

151views

Classify for which $M$ is $M+𝛼M^*$ diagonalizable for arbitrary $𝛼≠0$?

For which $M∈\text{Mat}(n×n, ℂ)$ is $M+𝛼M^*$ diagonalizable for arbitrary $𝛼∈ℂ \setminus \{0\}$? Alternatively just choose $𝛼∈ℝ \setminus \{0\}$ and analyse, when $M+𝛼M^\mathrm{T}$ is ...

Sinthoras

103

askedSep 17 at 10:29

Question 7

2votes

2answers

356views

Exponential of a Non-Triangular Matrix is Non-Triangular

I'd like to show that, given $A$ a non-upper triangular matrix, then $\exp(A)$ mustn't be upper triangular either. Equivalently, I could show that the logarithm of an upper triangular matrix is always ...

CallieWallie

107

askedSep 12 at 3:07

Question 8

0votes

3answers

82views

For a given matrix with real parameter $\alpha$, determine if there is an $\alpha$ such that $\beta(x, y)=y^T A_\alpha x$ is an inner product.

I've recently practiced some old linear algebra exams and came across this question.Given $\alpha \in \mathbb{R}$ and$$A_\alpha=\left[\begin{array}{lll}0 & 0 & \alpha \\ 0 & 1 & 0 \\...

Johann Kleindopf

27

askedSep 10 at 20:15

Question 9

0votes

1answer

79views

Diagonalization of an upper triangular matrix

I am considering a certain Lie algebra, in particular a complex upper-triangular Lie algerbra. Furthermore, I wish to find a nice way to write the exponential of an arbitrary element in this upper-...

CallieWallie

107

askedAug 27 at 4:56

Question 10

1vote

1answer

129views

Is this diagonalization formula incorrect?

Note: After posting I realized that it may be difficult to distinguish between $\mathfrak{x}_{\bar{i}}$ and $\mathfrak{x}_{\tilde{i}}$ when viewed with a web browser. I suggest using ...

Steven Thomas Hatton

3,859

askedAug 24 at 8:07

Question 11

0votes

1answer

74views

Efficiently computing the trace of products of diagonalizable matrices

Suppose that I have two matrices $A$ and $B$ which are both symmetric: $A=A^T, B=B^T$. Moreover, I know how to diagonalize both $A$ and $B$.Now I would like to define $T=A^{1/2}BA^{1/2}$, which is ...

miggle

391

askedAug 5 at 0:11

Question 12

1vote

1answer

70views

Alternative attempt to prove self-adjoint spectral theorem - can this work?

I've read the standard proof of the self-adjoint spectral theorem which uses induction, but I thought of another idea and am attempting to make it work, I'd like to know if I'm on to something or if ...

Zacharie Etienne

11

askedAug 1 at 3:54

Question 13

1vote

0answers

76views

Given a matrix of the form $M=[(A,B),(0,I)]$, show $M$ is diagonalizable iff $A$ is diagonalizable

This question in the sample final for my linear algebra class has me, for the first time in this class, stumped (and the prof hasn't released a solution set): Suppose $M$ is a matrix of the form $\...

Ryder Mendelson

325

askedJul 29 at 6:49

Question 14

0votes

2answers

82views

Extending a pair of left- and right-eigenvectors to a full eigenbasis

When a matrix is diagonalizable, it is possible to find left and right eigenvectors $u_i$ and $v_j$ such that $u_i^T v_j = \delta_{ij}$ (Kronecker delta), $u_i^T A = \lambda_i u_i^T$, and $A v_i = \...

Heraklit

331

askedJul 28 at 15:14

Question 15

0votes

0answers

33views

Coprime annihlating polynomials

Only the zero matrix admits two coprime annihilating polynomials. Is this true or false.I think it's true but the notebook I use claims it's false without elaborating.My approach is the following:...

Z23_0

1

askedJul 25 at 11:09

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Stack Exchange Network

Questions tagged [diagonalization]

Why is a real symmetric matrix diagonalizable?

Can one construct a liar's sentence using the diagonal lemma and the principle of explosion?

Inductive proof of necessary and sufficient conditions for diagonalisability

Definite unimodular form equivalent to a diagonal form while itself being non-diagonalizable

Showing that two $2\times 2$ matrices are similar

Classify for which $M$ is $M+𝛼M^*$ diagonalizable for arbitrary $𝛼≠0$?

Exponential of a Non-Triangular Matrix is Non-Triangular

For a given matrix with real parameter $\alpha$, determine if there is an $\alpha$ such that $\beta(x, y)=y^T A_\alpha x$ is an inner product.

Diagonalization of an upper triangular matrix

Is this diagonalization formula incorrect?

Efficiently computing the trace of products of diagonalizable matrices

Alternative attempt to prove self-adjoint spectral theorem - can this work?

Given a matrix of the form $M=[(A,B),(0,I)]$, show $M$ is diagonalizable iff $A$ is diagonalizable

Extending a pair of left- and right-eigenvectors to a full eigenbasis

Coprime annihlating polynomials

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