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.

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1answer
22 views

prove that matrix is diagonal by matrix rank and eigenvalue rank

$A$ is matrix $9\times9$ with rank of $5$, there is rank$(A-3I)=5$, the matrix has another eigenvalue of 5. I need to prove that $A$ is diagonal and find the similar diagonal matrix of $A$. I'm stuck, ...
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2answers
26 views

How to determine diagonalization [on hold]

Let $p(t) = t^2(t^2 + 4t + 4)(t - 5)^7$ be the characteristic polynomial of a matrix $M$. Suppose $\dim Null(M) = 2$ and $\dim Null (M - 5I) = 7$. Indicate which of the following statements is true: ...
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1answer
28 views

W is T-invariant. Define $\bar T: V/W \to V/W, \bar T(v+W)=T(v)+W$.Prove if $T_W$ and $\bar T$ are diagonalizable without common eigenvalue, then is T

$T$ is a linear operator on a finite dimensional vector space $V$, and $W$ be a $T$-invariant subspace of $V$. Define $\bar T: V/W\to V/W$ by $\bar T(v+W)=T(v)+W$. It can be proved that $\bar T$ is ...
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2answers
52 views

Compute power of a matrix $A$ as $n\rightarrow \infty$

We are given $A^p=A ...A$(p times) And we are given matrix A: $A=\begin{vmatrix}0.6&-0.4&0\\-0.4&0.6&0\\0&0&0.5\end{vmatrix}$ I need to compute $A^p$ as p approach Infinity. ...
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2answers
17 views

Special method of solution for $A\vec x=\vec b$ where $A$ is a square matrix such that $A^tA$ is diagonal and has full rank?

Is there any special shorter method of solution other than cramer's rule for solving a system of $n$ linear equations in $n$ unknowns $A\vec x=\vec b$ where the square matrix $A$ has the property that ...
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1answer
28 views

Roots of Matrices and Diagonalization

Question: For which of the following matrices $A_i$ is there A complex matrix $B$ such that $B^2 = A_i$; A self-adjoint complex matrix $B$ such that $B^2 = A_i$; A real matrix $B$ such that $B^2 = ...
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1answer
28 views

Prove that $A$ diagonalizable.

Let $A$ be an $n \times n$ matrix, and let $v_1,...,v_n$ be a basis of $R^n$ so that each $v_i$ is an eigenvector of $A$. Prove that $A$ diagonalizable. Does the diagonalization of $A = QDQ^{-1}$ ...
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3answers
234 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 ...
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2answers
35 views

For what values of k is this singular matrix diagonalizable?

So the matrix is the following: \begin{bmatrix} 1 &1 &k \\ 1&1 &k \\ 1&1 &k \end{bmatrix} I've found the eigan values which are $0$ with an algebraic multiplicity of $2$ ...
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0answers
35 views

Interpreting a diagonalized matrix

I'm doing a practice question before a test: The stress in a solid at a point P can be described by a matrix T called the stress matrix (or stress tensor). If n is a normal vector to a plane cutting ...
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0answers
44 views

Linear Algebra - tricky question regarding diagonalization

EDIT:: I was trying to figure out the matrix P for this question, but couldn't find connection between the eigenvalues (given in the diagonal matrix), the vectors given and matrix A. I've recently ...
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2answers
54 views

Matrix diagonalisable in R, but not in C.

I know is quite easy to find a matrix $A\in\mathbb{R}^{2,2}$ that is diagonalisable if the base field is $\mathbb{C}$, but not diagonalisable if the base field is $\mathbb{R}$. The easiest example can ...
0
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1answer
19 views

Finding eigenvalues/vectors of a matrix and proving it is not diagonalisable.

I have got the following matrix. $$\begin{pmatrix} -7 &4 \\ -9 &5 \end{pmatrix}$$ I need to find the eigenvalues, eigenvectors and $\textbf{prove}$ that it is not diagonalisable. I have ...
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3answers
203 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 ...
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1answer
71 views

Diagonalize tri-diagonal symmetric matrix

How to diagonalize the following matrix? \begin{pmatrix} 2 & -1 & 0 & 0 & 0 & \cdots \\ -1 & 2 & -1 & 0 & 0 & \cdots \\ 0 & -1 & 2 & -1 & 0 ...
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1answer
14 views

Symmetric matrix with with all positive/zero elements. How to ensure it is PSD?

I have the following matrix A: symmetric all positive and/or zero values the main diagonal is all the same variable, x. To ensure that the matrix A, is PSD, must I only ensure that x>=0? It seems ...
1
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1answer
25 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
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0answers
11 views

Compute the derivatives of an equation

I have an equation which is equal to: $(-c/2)ln(x) + (-c/2)tr(diag(B^TSB)x^{-1})$ Where $c$ is a constant, $tr$ represents the trace, $diag$ represents the diagonal. $B$, $S$ and $x$ are three ...
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4answers
71 views

Understanding the Jordan form via this example

The Question: I do not understand why the Jordan Form of the matrix: $A:=\begin{pmatrix} 1 & 1 \\ -1 & 3 \end{pmatrix}$ is: $J:=\begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$? Here is ...
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1answer
32 views

Derivation of matrix diagonalisation formula? [closed]

I can't seem to find a derivation of $A=P^TDP$ or an explanation of why this works or why it is important. I would be very grateful if someone could explain this, or perhaps give any useful lis about ...
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1answer
16 views

Proof if $A$ is normal then it is nondefective

What is the proof that if $A$ ($m\times m$ Matrix) is normal i.e $(AA^{\ast} = A^{\ast}A)$ then $A$ is non defective i.e (for each eigenvalue of $A$, its algebraic multiplicity is equal to the ...
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2answers
47 views

Diagonalize the $n \times n$ matrix with ones along both diagonals.

I'm having some trouble diagonalizing this nxn matrix with ones along both diagonals: $\begin{bmatrix} 1&0&0&\cdots&0&0&1\\0&1&0&\cdots&0&1&0\\ ...
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2answers
26 views

Find a and b such that the matrix is diagonalizable

Find a and b such that the matrix $$ \left( \begin{array}{ccc} 1 & a \\ 0 & b \\ \end{array} \right) $$ is diagonalizable. I know that $$ D = S^{-1} A S $$ where S is a matrix made of the ...
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1answer
25 views

Linear Algebra 3x3 matrix diagonalization Row operation before inversing

Hello I am diagonalizing the matrix $$\begin{bmatrix} -1 & 2 & 2 \\ 2 & 2 & -1 \\ 2 & -1 & 2 \end{bmatrix}.$$ The eigenvalues I found are $-3$ and $3$. The eigenvectors are ...
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2answers
55 views

Prove that the eigenvectors of this matrix are a basis in $\mathbb{R}^n$

Let $A \in \mathbb{R}^{n \times n}$ and $w \in \mathbb{R}^n$. Suppose that, $w_i>0$ and $a_{i,j} = w_i / w_j$ for all $i,j=1,\dots,n$. Note that from the construction comes that ...
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1answer
24 views

Prove that the columns of the similarity matrix of a diagonalization are the eigenvectors

I'm interested in eigendecomposition of a matrix. It is clear for me, that you can eigendecompose a matrix if and only if it is diagonalizable. I'm looking for a short proof for that statement, that ...
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1answer
32 views

Three Simultaneously Diagonalizable Matrices

I have three symmetric square matrices $M$, $G$, and $S$ with the following properties: $S$: symmetric and positive semi-definite. $M$: Fully diagonal with positive entries. $G$: is a subset of ...
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2answers
28 views

Similar matrices C and D: how to derive the relation $\mathbf{x} = S^{-1} \mathbf{y}$ when $C = S^{-1}DS$

D (with corresponding eigenvector $\mathbf{x}$) and C (with corresponding $\mathbf{y}$) are similar matrices, which means they have the same eigenvalues. So the relation $C = S^{-1}DS$ holds. So we ...
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4answers
253 views

Exercise about Matrix diagonalization

Well I have to diagonalize this matrix : $$ \begin{pmatrix} 5 & 0 & -1 \\ 1 & 4 & -1 \\ -1 & 0 & 5 \end{pmatrix} $$ I find the polynome witch is $P=-(\lambda-4)^2(\lambda-6)$ ...
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2answers
43 views

Prove a matrix is not diagonalizable

To show that a matrix is not diagonalizable, I would just have to show that there are no eigenvalues present in the matrix. So, for example, if I want to prove that $$A=\begin{bmatrix} 0 & -1 ...
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1answer
49 views

Shortcut finding D when diagonalizing a matrix when encountered with a Householder reflection

P is given as P = $\left(\begin{array}{rrr} 1 & 1 & 1\\ 1 & 0 & -2\\ 1 & -1 & 1 \end{array}\right).$ It is known that P is invertible. I is a 3x3 identity matrix Supposed ...
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2answers
32 views

Show a matrix is similar to a lower triangular matrix

$A = \left(\begin{array}{cc}2 & -1 \\0 & 2\end{array}\right)$ $B = \left(\begin{array}{cc}\lambda & 0 \\1 & \lambda\end{array}\right)$. I know that the $\lambda = 2$. And $r(1,0)^t$, ...
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2answers
33 views

Finding a non-singular matrix $C$ such that $C^{-1}AC$ is diagonal

$A = \left(\begin{array}{cc}1 & 0 \\1 & 3\end{array}\right)$. I find the eigenvalues = 1,3. The eigenvector corresponding to 1 = $t(1,-2)^t$. The eigenvector corresponding to 3 = $r(1,0)$. ...
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0answers
22 views

Find a matrix $P$ for a square matrix $B$ with all entries $(B)_{ij} = b$, $b \in R$. $P$ is a matrix that orthogonally diagonalize matrix $B$.

The condition must meet $$D = (P^{T})BP $$ or $$D=(P^{-1})BP$$ I'm having trouble finding a pattern for all entries and infinite square size matrix. I found a matrix P that is 5x5 with all ...
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1answer
27 views

Diagonalising an infinite-dimensional Hermitian square matrix

I have a quantum state which takes the following form: $$\rho = \sum_{b, \,c \, = \,0}^\infty \frac{(-igt)^b(igt)^c}{\sqrt{b!c!}}\vert b\rangle\langle c\vert.$$ This is an infinite Hermitian matrix ...
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0answers
11 views

Dimensions of matrices around a diagonal matrix?

The matrices $\mathbf{L}$, $\beta$ and $\mathbf{c}$ are ($j \times b$), ($b \times 1$) and ($j \times 1$) dimensional, respectively, with $j \le b$. The matrix $\mathbf{X}' \mathbf{X}$ is a diagonal ...
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1answer
50 views

Understanding Jordan Canonical Form.

Two questions: How does the nilpotent index $k$ of a linear transformation L on a vector space of dimension $n$ relate to possible Jordan Canonical Forms? My understanding is that a Jordan block ...
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0answers
33 views

Does gaussian elimination always work?

If so, why don't we use that to get from any square matrix to a triangular matrix - from which can be deduced eigenvalues, determinant (product of eigenvalues) and diagonal matrix (since the diagonal ...
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1answer
30 views

Diagonalization: How to show that A exists $S^2 = D$ given that D is a nonegative diagonal marix

(a) Show that if D is a diagonal matrix with nonnegative entries on the main diagonal, then there is a matrix S such that $S^2 =D$ SOLVED (b) Show that if A is a diagonalizable matrix with ...
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0answers
20 views

How to Solve: Find bases that Diagonalize a Matrix

Given bases, $w_1 = (1,0)$ and $w_2 = (0,1)$. Find bases $(e_1,e_2,e_3)$ in $R^3$ relative to which the matrix of $T$ = $ \left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & ...
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2answers
78 views

If $A^3 = A$ then the eigen values are all 1 right?

Since $A^n = PD^nP^{-1}$ where D is a matrix consisting only of the eigenvalues of on its leading diagonal. For the scenario to be true $D^B = D$ which is only true if the eigenvalues are all 1s ...
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2answers
38 views

Diagonalization: Am i finding these eigenvectors wrongly?

$$A=\begin{bmatrix} 1&-2&-8\\ 0&-1&0\\ 0&0&-1 \end{bmatrix}$$ $$P=\begin{bmatrix} 1&-4&1\\ 1&0&0\\ 0&1&0 \end{bmatrix}$$ Confirm that P diagonalizes A. ...
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3answers
49 views

Using factorisation, $A=PJP^-1$ to compute $A^k$

Using factorisation, $A=PJP^{-1}$ to compute $A^k$, where $k$ represents an arbitrary positive integer. $$ \begin{bmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{-1} & \mathbf{2} \end{bmatrix} = ...
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2answers
34 views

A,B are diagonalizable matrix and their characteristic polynomials are the same.prove that $A$ and $B$ are similar

let A,B are diagonalizable matrix in ${c^n}$ and their characteristic polynomials are the same. can we prove that $A$ and $B$ are similar?
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2answers
93 views

Conditions for a Matrix to be Diagonalizable

Let $M$ be a matrix with the entries $a_{1}, ..., a_{n}$ on the secondary diagonal (the one that ranges from $m_{n1}$ to $m_{1n}$) with all other entries being $0$. Find under which conditions the ...
2
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2answers
151 views

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable. I know how to find the eigenvalues, and diagonalizing ...
0
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3answers
72 views

Matrix diagonalization - eigenvalues on diagonal

Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $\Delta$ such that $A=PD P^{-1}$ where $D$ is a diagonal matrix. What theorem tells us that $P$ is a matrix composed of the ...
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4answers
162 views

Showing when a permutation matrix is diagonizable over $\mathbb R$ and over $\mathbb C$

For a permutation $\sigma$ of the set $\{1,...,n\}$, and consider the $n \times n$ matrix $A_\sigma$, where the $i^{\text{th}}$ column is the standard vector $e_{\sigma (i)}$. For which $\sigma$ is ...
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2answers
45 views

powers of a diagonal matrix to infinity

Let $A$ be a square matrix that is diagonalizable. This means that it can be like this: $A = SDS^{-1}$, where $D$ is a diagonal matrix containing the eigenvalues of $A$. It follows $S$ contains the ...
2
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2answers
85 views

Let $A$ be a complex matrix such that $A^n = I$, show that $A$ is diagonalisable.

Let $A$ be a complex matrix such that $A^n = I$, show that $A$ is diagonalisable. How do I do this? I would hazard a guess that since $A^n = I$ that means that $A^n$ is obvious diagonal, and I ...