# Questions tagged [diagonalization]

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.

1,459 questions
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### Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition?

I'm trying to intuitively understand the difference between SVD and eigendecomposition. From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three ...
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### Eigenvalues of the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
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### Are most matrices diagonalizable?

More precisely, does the set of non-diagonalizable (over $\mathbb C$) matrices have Lebesgue measure zero in $\mathbb R^{n\times n}$ or $\mathbb C^{n\times n}$? Intuitively, I would think yes, since ...
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### Is every self-inverse matrix diagonalizable?

If $A=A^{-1}$, is there always a matrix C such that $C^{-1}AC$ is a diagonal matrix (containing only -1 and 1 in the main diagonal) ? How can I check with PARI/GP, if a given matrix is ...
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### An operator that commutes with another operator $T$ with distinct characteristic values is a polynomial in $T$

I'm trying to solve some problems in Hoffman and Kunze and I'm kind of stuck on this one. This is 6.5.3 on Hoffman and Kunze. Here is the question: Let $T$ be a a linear operator on an $n$-...
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### How many non-diagonalizable $2\times 2$ matrices are there with all entries single-digit strictly positive integers?

I'd like to know how many non-diagonalizable size 2 matrices there are with integer coefficient between 1 and 9. I built a python program which counts the non-diagonalizable matrices with such ...
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### Diagonalizable properties of triangular matrix

How to show that an upper triangular matrix with identical diagonal entries is diagonalizable iff it is already diagonal?
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### Find real number $a$ such that matrix $A$ is NOT diagonalisable

Consider the matrix: $\begin{bmatrix}2 & a & -1\\0 & 2 & 1\\-1 & 8 & -1\end{bmatrix}$ Find all $a \in \mathbb R$ such that $A$ is not diagonalisable. I've never thought of ...
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### How prove this linear algebra $AB=BA$?

Suppose $A,B\in M_{n}(\Bbb C)$ satisfies for $\forall a,b\in \Bbb C,aA+bB$ is always diagonalizable. Show that $$AB=BA.$$
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### Quick way to check if a matrix is diagonalizable.

Is there any quick way to check whether a matrix is diagonalizable or not? In exam if a question is asked like "Which of the following matrix is diagonalizable?" and four options are given then how ...
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### Prove that matrix $A$ diagonalizable if $A^2=I$ using characteristic polynomial

Prove that the matrix $A$ is diagonalizable if $A^2=I$ using characteristic polynomial I saw an answer that used the minimal polynomial of $A$. Can that be proven without using minimal polynomial? ...
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### When a matrix has same eigenvalues of its column-swapped version?

What are the properties needed for a matrix $A$ to have $\mbox{Spec}(A)= \mbox{Spec}(A \cdot P)$, where P = \begin{pmatrix} 0 & \cdots & 0 & 1 \\ \vdots & \...
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### Doubts about a question I asked a long time ago (eigenvalues)

Here I posted a question about the eigenvalues of the matrix $A:=vv^t$ (where $v\in\mathbb{R}^n$). The question was answered but I think (after some time) that I am not satisfied. Can someone ...
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### Motivation behind matrix diagonalisation

I'm going to give a 50 minutes lecture about matrix diagonalization for first year college students and I would like to give some applications of it. I've been thinking about saying the calculation of ...
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### If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
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### Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.

Let $A$ be an $n \times n$ complex matrix. Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ such that a. A = D + N b. DN = ND and show that these matrices are uniquely ...
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### Diagonalizing Quadratic Forms. Linear Algebra

I have a question that reads: Diagonalize the quadratic form $A(x,y) = 3x^2 -12xy + 7y^2$ by completing the square. What is diagonalization? Is that when I should find the eigenvector matrix, say,...
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### Is a matrix diagonalizable, if one of its eigenvalues is zero?

I checked weather the following matrix is diagonalizable. $$A=\begin{bmatrix} 4 & 0 & 4\\ 0 & 4 & 4\\ 4 & 4 & 8 \end{bmatrix}$$ And the corresponding eigenvalues were $0$, $4$...
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### $M^2=I_n$ implies $M$ diagonalizable

Let $M$ be an $n \times n$ matrix such that $M^{2}=I_n$; does this imply that $M$ is similar to a diagonal matrix $D$? How can we prove this?
I have not seen anywhere written that it is impossible, but it seems impossible, so I want to check if I missed something. According to a theorem, an $n\times n$ matrix is diagonalizable if it has \$...