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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.

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 $T$-...
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 ...
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 ...
4
votes
6answers
34k views

Finding a 2x2 Matrix raised to the power of 1000

Let $A= \pmatrix{1&4\\ 3&2}$. Find $A^{1000}$. Does this problem have to do with eigenvalues or is there another formula that is specific to 2x2 matrices?
5
votes
2answers
6k views

$AB=BA$ with same eigenvector matrix

I read in G. Strang's Linear Algebra and its Applications that, if $A$ and $B$ are diagonalisable matrices of the form such that $AB=BA$, then their eigenvector matrices $S_1$ and $S_2$ (such that $A=...
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?
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 $A\in\mathbb{C}^n$ ...
1
vote
3answers
154 views

Writing an expression as a sum of squares

I'd like to write $2xy+2xz+2yz$ in the form $a(\cdots)^2+b(\cdots)^2+c(\cdots)^2$ where each blank space is a linear combination of $x,y,z$. The closest I have is: $$(x+y+z)^2-(x-z)^2-y^2=2xy+4xz+2yz$...
2
votes
2answers
400 views

Simultaneous Diagonalization of two bilinear forms

I need to diagonalize this two bilinear forms in the same basis (such that $f=I$ and $g$=diagonal matrix): $f(x,y,z)=x^2+y^2+z^2+xy-yz $ $g(x,y,z)=y^2-4xy+8xz+4yz$ I know that it is possible ...
3
votes
2answers
3k views

Finding $P$ such that $P^TAP$ is a diagonal matrix

Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in M_n(\mathbb{C})$$ Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix. So here's the solution: $$A = \...
2
votes
3answers
300 views

diagonalisability of matrix few properties

What are the most important things one should remember to check the diagonalisability of a matrix? Please help, I have exams on next week. Say some best and easy methods,time efficient.
4
votes
2answers
3k views

Minimal polynomial of diagonalizable matrix

Prove that a matrix $A$ over $\mathbb{C}$ is diagonalizable if and only if its minimal polynomial's roots are all of algebraic multiplicity one.
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 ...
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 ...
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 ...
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?
1
vote
1answer
278 views

Is there any connection between a matrix being invertible and being diagonalizable?

Let $A$ be an $n\times n$ matrix. If $A$ is invertible then one of it's eigenvalues is 0. If $A$ is diagonalizable then it has $n$ linearly independent eigenvectors. Are these two statements true? ...
0
votes
2answers
393 views

How to diagonalize $f(x,y,z)=xy+yz+xz$

Could you tell me how to diagonalize $f(x,y,z)=xy+yz+xz$. I know I can rewrite it as $(x+ \frac{1}{2}y + \frac{1}{2}z)^2 - x^2 - \frac{1}{4}(y-z)^2$ What do I do next? Could you help me?
4
votes
3answers
2k views

If $A^2 = I$, then $A$ is diagonalizable, and is $I$ if $1$ is its only eigenvalue

Let $A$ be a square matrix of order $n$ such that $A^2 = I$. Prove that if $1$ is the only eigenvalue of $A$, then $A = I$. Prove that $A$ is diagonalizable. For (1), I know that there ...
4
votes
1answer
409 views

Proving a diagonal matrix exists for linear operators with complemented invariant subspaces

I came across this problem one of my practice worksheets and I was stumped as to how I would go about solving this. Let $T : V \rightarrow V$ be a linear operator on a finite dimensional vector space ...
1
vote
0answers
163 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
0
votes
1answer
312 views

Simultaneous diagonlisation of two quadratic forms, one of which is positive definite

Let $\varphi, \phi$ be quadratic forms on $V$ and suppose $\varphi$ is positive definite. I want to find a basis for V such that $\varphi$ and $\phi$ are both represented by diagonal matrices. My ...
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 ...
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 : ...
5
votes
3answers
222 views

$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$

$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$, consider the linear map $T:M_2(\mathbb{R})\to M_2(\mathbb{R}):=B\to AB$ Then which of the following are true? $T$ is ...
4
votes
4answers
3k views

Diagonalizable matrices that commute share eigenspace

I know it's been answered before (at least to the case with $n$ different eigenvalues) but I didn't find a proof for the general case, and I would like some help with this question. We are given ...
4
votes
1answer
886 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is that,...
4
votes
2answers
3k views

diagonalize quadratic form

I have this quadratic form $Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$ And they ask me: for which values of $x,y$ and $z$ is $Q=0$? and I have to diagonalize also the quadratic form. I calculated ...
4
votes
3answers
251 views

Diagonalize a symmetric matrix

let $$A = \left(\begin{array}{cccc} 1&2&3\\2&3&4\\3&4&5 \end{array}\right)$$ I need to find an invertible matrix $P$ such that $P^tAP$ is a diagonal matrix and it's main ...
2
votes
1answer
81 views

Let $A$ and $B$ be matrices over $\mathbb C$. Then pick out the correct statements.

Let $A$ and $B$ be matrices over $\mathbb C$. Then, $AB$ and $BA$ always have the same set of eigenvalues. If $AB$ and $BA$ have the same set of eigenvalues then $AB=BA$. ...
1
vote
1answer
349 views

The diagonalizable matrices are not dense in the square real matrices

Suppose that $n \ge 2$. How to prove that the set $\mathcal D \subset M_n(\mathbb R)$ of the diagonalizable real matrices is not dense in $M_n(\mathbb R)$?
2
votes
2answers
370 views

$A$ be a complex $3\times 3$ matrix such that $A^3=-I$

Let $A$ be a complex $3\times 3$ matrix such that $A^3=-I$, then we need to find out which of the following statements are correct? $A$ has three distinct eigenvalues; $A$ is diagonalizable over $\...
2
votes
2answers
571 views

Diagonalization of an infinite matrix

Let A be an infinite matrix with all its first column elements equal to 1 and the rest of them equal to 0. A=\begin{pmatrix} 1 & 0 & 0 & 0 & \cdots\\ 1 & 0 & 0 & 0 &\...
2
votes
3answers
4k views

New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
2
votes
2answers
931 views

If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable

Suppose that $A \in M_{n\times n}(\Bbb F)$ has two distinct eigenvalues $\lambda_{1}$ and $\lambda_{2}$ and that $\dim (E_{\lambda_1})=n-1$ show that $A$ is diagnolizable.
0
votes
4answers
275 views

Is the matrix diagonalizable for all values of t?

For t∈R, let $A_t = \left( \begin{array}{ccc} t & 1 & 1 \\ 1 & t & 1 \\ 1 & 1 & t \end{array} \right) $. Find the Eigenvalues and Eigenvectors. Is $A_t$ diagonalizable for all ...
6
votes
3answers
4k views

Where M is a matrix calculate a formula for M^n

Let $$M = \begin{bmatrix} -7 & 8 \\ -8 & -7 \end{bmatrix}.$$ Find formulas for the entries of $M^n$ where $n$ is a positive integer. (Your formulas should not contain complex numbers.) Your ...
4
votes
1answer
282 views

Diagonalize the matrix A or explain why it can't be diagonalized

Diagonalize the matrix or explain why it cant be diagonalized $A=\begin{pmatrix}1 & 2 & 4 \\3 & 5 & 2 \\2 & 6 & 1\end{pmatrix}$ Hint: One eigenvalue is $λ=9$ So, i began the ...
3
votes
1answer
143 views

Two questions about diagonalization

Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
2
votes
2answers
109 views

Proving that if $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$, then $A$ is diagonalizable.

If $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$ then prove that $A$ is diagonalizable. I have got that the minimal polynomial of $A$ may be $x^3-x$ or $x$ or $x(x+1)$ or $x(x-1)$ in the ...
1
vote
3answers
102 views

Finding Matrix A from Eigenvalues and Eigenvectors (Diagonalization)

Question: Let $A$ be a $3 \times 3$ Matrix such that $[-3,4,1]$ is the eigenvector corresponding to eigenvalue $3$, and $[6,-3,2]$ is an eigenvector corresponding to the eigenvalue $2$. If $v$ = $...
1
vote
3answers
164 views

Determining $\mbox{tr}(A)$ and $\det(A)$ as functions of the eigenvalues of a matrix

So if I have an $n\times n$ matrix $A$ that is diagonalizable then how do I determine $\mbox{tr}(A)$ and $\det(A)$ as functions of the eigenvalues of the matrix? For $\det(A)$, I know the formula ...
1
vote
2answers
41 views

How to check if a matrix is diagonizable?

So i have this $3\times 3$ matrix $$ \begin{pmatrix} -2 & 0 & 0 \\ 3 & 1 & -6 \\ 0 & 0 & -2 \\ \end{pmatrix} $$ I want to check if the matrix is ...
1
vote
1answer
627 views

Conditions for diagonalizability of $n\times n$ anti-diagonal matrices

Let $A$ be an $n\times n$ anti-diagonal matrix: $a_{i,j}=0$ unless $i+j=n+1$. A) When is $A$ diagonalizable (what are the conditions on the $a_{i,n+1−i}$)? B) Find the eigenvalues and eigenvectors ...
1
vote
1answer
340 views

Find a matrix that simultaneously diagonalizes to matrices

struggling with a question from homework and would appreciate some assistance. Let $A, B \in M_2^{\mathbb{R}}$ be defined as follows: $$A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix},...
0
votes
2answers
68 views

Diagonalisable or not? [closed]

Let $$A = \begin{pmatrix}a & b\\c & d\end{pmatrix}.$$ Show that 1) $A$ is diagonalisable if $(a - d)^2 + 4 bc > 0$ 2) $A$ is not diagonalisable if $(a - d)^2 + 4 bc < 0$
0
votes
1answer
70 views

Digonalizable Polynomial Matrix

For any polynomial $p(x) = a_0 + a_1x + · · · + a^kx^k$ and any square matrix A, $p(A)$ is defined as $p(A) = a_0I + a_1A + · · · + a_kA^k$. Show that if v is any eigenvector of A and $χ_A(x)$ is the ...
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 ...
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 ...
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?