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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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 ...
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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 $T$ ...
3
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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 ...
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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):
$$
\left(\begin{array}{c|c}
1 & 0\\\hline
0 &...
2
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2
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Showing that two 2ドル\times 2$ matrices are similar
I want to prove that the following 2ドル\times 2$ matrices are similar over $\mathbb R$:
$$
A=\begin{pmatrix}
0 & -1\\
1 & 0\\
\end{pmatrix}
\qquad
B=\begin{pmatrix}
-3 & -5\\
2 & 3\\
\...
2
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1
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Classify for which $M$ is $M+𝛼M^*$ diagonalizable for arbitrary $𝛼≠0$?
For which $M∈\text{Mat}(n×ばつn, C)$ 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 ...
2
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2
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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 ...
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3
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For a given matrix with real parameter $\alpha,ドル 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 \\...
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1
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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-...
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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 ...
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1
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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},ドル which is ...
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1
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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 ...
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Given a matrix of the form $M=[(A,B),(0,I)],ドル 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 $\...
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2
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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,ドル and $A v_i = \...
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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:
...