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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.

88
votes
7answers
38k views

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 ...
36
votes
3answers
20k views

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 ...
36
votes
2answers
3k views

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 ...
28
votes
5answers
6k views

If two matrices have the same eigenvalues and eigenvectors are they equal?

The question stems from a problem i stumbled upon while working with eigenvalues. Asking to explain why $A^{100}$ is close to $A^\infty$ $$A= \left[ \begin{array}{cc} .6 & .2 \\ .4 & ...
26
votes
2answers
27k views

Simultaneous diagonalization

Let $V$ be a vector space of finite dimension and let $T,S$ linear diagonalizable transformations from $V$ to itself. I need to prove that if $TS=ST$ every eigenspace $V_\lambda$ of $S$ is $T$-...
20
votes
5answers
14k views

Symmetric matrix is always diagonalizable?

I'm reading my linear algebra textbook and there are two sentences that make me confused. (1) Symmetric matrix $A$ can be factored into $A=Q\lambda Q^{T}$ where $Q$ is orthogonal matrix : ...
19
votes
3answers
4k views

Block Diagonal Matrix Diagonalizable

I am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diagonalizable, if only if $A$ and $B$ are diagonalizable. If $A\in\mathbb{C}^n$ ...
14
votes
5answers
922 views

Find large power of a non-diagonalisable matrix

If $A = \begin{bmatrix}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$, then find $A^{30}$. The problem here is that it has only two eigenvectors, $\begin{bmatrix}0\\1\\1\...
13
votes
2answers
14k views

Can a matrix be invertible but not diagonalizable? [duplicate]

While reading a chapter on diagonalizable matrices, I found myself wondering: Can a matrix $A \in \mathbb R^{n \times n}$ be invertible but not diagonalizable? My quick Google search did not ...
13
votes
2answers
410 views

Diagonalization: Can you spot a trick to avoid tedious computation?

I am studying for my graduate qualifying exam and unfortunately for me I have spent the last two years studying commutative algebra and algebraic geometry, and the qualifying exam is entirely '...
12
votes
3answers
359 views

Suppose $e^A = A$, prove that $A$ is diagonalizable

Suppose $e^A = A$, prove that $A$ is diagonalizable, where A is a matrix. What I have tried to do is write $A= D + N$, where $D$ is diagonalizable, $N$ is nilpotent and $DN = ND$. Since $N$ is ...
12
votes
1answer
395 views

Proof of a theorem on simultaneous diagonalization from Hoffman and Kunze.

Now I am reading Linear Algebra from the book of Hoffman and Kunze second edition. I am trying to understand theorem $8$ on pg number $207$ which is based on Simultaneous diagonalization. I have seen ...
12
votes
2answers
483 views

Let $S$ be a diagonalizable matrix and $S+5T=I$. Then prove that $T$ is also diagonalizable.

My solution: Since $S$ is diagonalizable, so we can write $S=P^{-1}DP$, where $P$ is an invertible matrix and $D$ is a diagonal matrix. Now $5T=I-S=P^{-1}P-P^{-1}DP=P^{-1}(I-D)P$. So $T=P^{-1}\...
12
votes
0answers
464 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
11
votes
3answers
6k views

Diagonalization of a projection

If I have a projection $T$ on a finite dimensional vector space $V$, how do I show that $T$ is diagonalizable?
11
votes
2answers
2k views

Show that if $A^{n}=I$ then $A$ is diagonalizable.

Suppose $A$ is an $m \times m$ matrix which satisfies $A^{n}=1$ for some $n$, then why is $A$ necessarily diagonalizable. Not sure if this is helpful, but here's my thinking so far: We know that $A$ ...
11
votes
1answer
13k views

Hermitian Matrices are Diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...
10
votes
3answers
2k views

Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with ...
10
votes
3answers
14k views

Square root of Positive Definite Matrix

Let $A$ be an $n\times n$ positive definite matrix. Show that there exists a unique positive definite matrix $B$ such that $B^2=A$. I do know the existence. But what about the uniqueness? Would you ...
9
votes
3answers
890 views

Criterion for deciding whether matrix is diagonalizable

Let $B \in$ GL$_n(\mathbb{C})$. In a paper I'm reading someone probably claims the following: Lemma: For showing that $B$ is diagonalizable it suffices to show the following: Let $\lambda$ be an ...
9
votes
2answers
3k views

$A$ is some fixed matrix. Let $U(B)=AB-BA$. If $A$ is diagonalizable then so is $U$?

This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze. Here is the question. Let $V$ be the space of $n\times n$ matrices over a ...
9
votes
3answers
6k views

What's so useful about diagonalizing a matrix?

I'm told the the purpose of diagonalisation is to bring the matrix in a 'nice' form that allows one to quickly compute with it. However in writing the matrix in this nice diagonal form you have to ...
9
votes
2answers
458 views

trace , determinant and which of the following are true(NBHM-$2014$)

Let $A \in M_2(\mathbb R)$ be a matrix which is not a diagonal matrix . Which of the following statements are true?? a. If $tr(A)=-1$ and $detA=1$, then $A^3=I$. b. If $A^3=I$, then $tr(A)=-...
9
votes
1answer
8k views

Eigenvalues of outer product matrix of two N-dimensional vectors

I have a vector $\textbf{a}=(a_1, a_2, ....)$, and the outer product $M_{ij}=a_i a_j$. What are the eigenvalues of this matrix? and what can you say about the co-ordinate system in which $M$ is ...
9
votes
1answer
122 views

When can two matrices have zero diagonal in the same basis?

It's common to ask if two Hermitian matrices $A$ and $B$ can be diagonalized in the same basis. Is there an efficient way to check if they can be made hollow in the same basis? By hollow, I mean that ...
9
votes
1answer
169 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ \...
8
votes
4answers
3k views

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 ...
8
votes
2answers
2k views

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 $n$-...
8
votes
4answers
118 views

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 ...
8
votes
3answers
4k views

Diagonalizable properties of triangular matrix

How to show that an upper triangular matrix with identical diagonal entries is diagonalizable iff it is already diagonal?
8
votes
3answers
195 views

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 ...
8
votes
1answer
743 views

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 $$AB=BA.$$
8
votes
2answers
14k views

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 ...
8
votes
1answer
346 views

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? ...
8
votes
0answers
234 views

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 \begin{equation} P = \begin{pmatrix} 0 & \cdots & 0 & 1 \\ \vdots & \...
7
votes
6answers
1k views

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 ...
7
votes
7answers
765 views

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 ...
7
votes
1answer
2k views

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?
7
votes
3answers
2k views

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 ...
7
votes
2answers
4k views

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,...
7
votes
1answer
57 views

Matrix Representation of linear operators

Let $S:\mathbb R^n \to \mathbb R^n$ be given by $S(v) = \alpha (v)$, for a fixed $\alpha \in \mathbb R, \alpha \neq 0$. Let $T:\mathbb R^n \to \mathbb R^n$ be a linear operator such that $B = (v_1, ...
7
votes
1answer
1k views

Solving a recurrence with diagonalization?

Considering the recurrence $F_n=F_{n-1}+3F_{n-2}-3F_{n-3}$ where $F_0=0$, $F_1=1$ and $F_2=2$. Use diagonalization to find a closed form expression for $F_n$. So I first continued the recurrence to ...
7
votes
1answer
111 views

Eigenvalues of a $4\times 4$ parameters matrix

Let $a,b,c,d\in\Bbb{C}$ and $B =\begin{bmatrix} a & b & c & d\\ d & a & b & c\\ c & d & a & b\\ b & c & d & a\\ \end{bmatrix}$ I ...
6
votes
3answers
6k views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $X$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
6
votes
4answers
623 views

Show that this matrix is not diagonalizable

Say I have a matrix: $$A = \begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix} $$ Is this matrix diagonalizable? Does a 2x2 matrix always have 2 eigenvalues (multipicity counts). Why is this? I ...
6
votes
4answers
2k views

Why is $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ not diagonalizable

I have a number of sufficient conditions as to when a matrix $A$ is diagonalizable, namely: When $A$ is symmetric When $A$ has distinct eigenvalues Given $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 ...
6
votes
3answers
5k views

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 $0$, $4$...
6
votes
4answers
202 views

$M^2=I_n$ 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?
6
votes
2answers
8k views

Is it possible to diagonalize a singular matrix?

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 $...
6
votes
2answers
22k views

Proving that a symmetric matrix is positive definite iff all eigenvalues are positive

This has essentially been asked before here but I guess I need 50 reputation to comment. Also, here I have some questions of my own. My Proof outline: (forward direction/Necessary direction): Call ...