# Tagged Questions

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### Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
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### Eigenvector Proof $(I+A)^{-1}$.

Show that the eigenvectors of the $n \times n$ matrix A are also eigenvectors of the matrix $$M = (I+A)^{-1}$$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues. My Work: ...
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### Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector

Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries. The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, ...
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### Solve a System with Variable

Given these matrices, how does one find two real solutions? $dx/dt$ = $\begin{bmatrix} 3 & -5\\ 5 & 3 \end{bmatrix}x$ with $x(0) = \begin{bmatrix} 2\\ -3 \end{bmatrix}$
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### Common eigenvector of two linear transformation matrices

I have two linear transformation matrices \begin{pmatrix} 3 & 2 \\ -2 & 1 \end{pmatrix} and \begin{pmatrix} 1-a & -a \\ a & 1 \end{pmatrix} How to find out what the value of ...
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### Hermitian matrices [duplicate]

Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such that $\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H. How to prove ...
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### Hermitian matrix properties

Suppose we have hermitian matrix $H$, matrix $A$, composed of eigenvectors of $H$, such that $\langle A\mathbf i\mid A\mathbf i\rangle=1$, where $\mathbf i$ is the $i$-th column of matrix $H$. How ...
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### Delocalization of eigenvectors in Expanding Graphs

Given an adjacency matrix A, can we say something about whether the eigenvectors corresponding to its highest(or second-highest) eigenvalues are delocalized ? By delocalization I mean that every ...
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### How to prove that the corresponding matrix is unitary

Let's say we are given hermitian matrix $H$. How to prove that the matrix $M$, formed from eigenvectors of $H$ is unitary? Thanks
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### Dimension of the corresponding eigenspace?

I'm studying for my linear exam and would appreciate any help for this practise question: You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace? A = ...
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### A Has characteristic polynomial that can be reduced to linear products $\Rightarrow$ A similar to upper triangular Matrix

Prove that if $A\in M_{n}\left(\mathbb{F}\right)$ matrix with a characteristic polynomial that can be written as a product of linear elements (?) ...
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### Finding $A^n$ in terms of $P$ and $D$ (diagonalized)

My question is regarding the last two parts. I have Found $D$ and $P$, how can I obtain $A^{200}$ and det $(A^{200})$ form $PDP^{-1}?$ Thanks!
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### Eigenvectors and Kernel of Matrix

I'm trying to take find the eigenvectors of the matrix $$\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$ I've found the eigenvalues of $1$ and $0$. I'm ...
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### The connection between the rank of a matrix and its zero-mode eigenvectors

I would be most thankful if you could help me prove that if an arbitrary n by n matrix has rank m < n, then the matrix has (n-m) linearly independent eigenvectors corresponding to the eigenvalue ...
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### Limit of $A^k b$ for $k$ large

My textbook references the fact that for any matrix $A$ and any non-zero $b$ we have $$\lim_{k\rightarrow \infty} A^k b = v$$ Where $v$ is the eigenvector of $A$ with the largest eigenvalue. The proof ...
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### How do I write this matrix in Jordan-Normal Form

I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors ...
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### Diagonalizable Matrix $A^2$

How can I find a matrix $A$ such that $A^2$ is diagonalizable but $A$ is not? I have tried many different ways, but to no avail. Is there something that I am missing in the question that gives a ...
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### Expressing a matrix as an expansion of its eigenvalues

This shouldn't be too difficult but I can't find a satisfactory proof. Show that a real, symmetric matrix A satisfying the eigenvector equation $Au_{i} = \lambda u_{i}$ cam be expressed as an ...
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### How to calculate left and right eigenvector corresponding to the zero eigenvalue.

I'm working on $8\times8$ matrix resulting from the Jacobian of $8$ differential equation of a disease model evaluated at disease free equilibrium. I needed to get the left and right eigenvectors ...
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### What is the difference between first and second right eigenvectors of a row stochastic matrix and their meaning?

In an $n\times n$ non negative row stochastic matrix (rows sum up to 1). The entries of the stochastic matrix I have represent directed links between countries. Why is the first right eigenvector a ...
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### eigenvalues of the sum of a matrix with known eigenvalues and a diagonal matrix

Suppose $B = A+D$, where all the eigenvalues of $A$ are already known and $D$ is a diagonal matrix, how to compute the eigenvalues of B without diagonalizing $B$ directly?
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### trace of the inverse of a matrix times another matrix

I do know generally $\text{trace}(A^{-1}B)\not= \sum_i \lambda_{B_i}/\lambda_{A_i}$, where $\lambda_{A_i}$ and $\lambda_{B_i}$ are the corresponding eigenvalues of matrix $A$ and $B$ respectively, ...
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### eigenvalues of block matrix with the eigenvalues of one block already known

Give a matrix which can be decomposed into 4 parts $B = \left[\begin{matrix}A &I \\ -I &0\end{matrix}\right]$ where $I$ denotes the identity matrix and $0$ is a zero matrix. It's easy to ...
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### Eigenvalues and Eigenvectors of Large Matrix

Computing eigenvalues and eigenvectors of a $2\times2$ matrix is easy by solving the characteristic equation. However, things get complicated if the matrix is larger. Let's assume I have this matrix ...
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### Construct a matrix transform

consider $\frac{dx}{dt} = Ax$ where $A$ is the matrix $$\begin{bmatrix} 1 & 0 & 1 \\ 0 & 0 & -2 \\ 0 & 1 & 0 \\ \end{bmatrix}$$ ...
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### Question on generalized eigenspaces of commuting matrices

The following question came up as a though while I was reading. I cannot see how to proceed on it. Let us have $M_1,\ldots,M_n$ be commuting matrices. I know that that the generalized eigenspaces are ...
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### Can we always recover a matrix from its eigenvalues and eigenvectors?

If we're given all the eigenvalues of a square matrix $A$ and the corresponding eigenvectors of each eigenvalue, then in what case(s) is it possible theoretically to recover $A$ from this much ...
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### What are the eigenvectors of this normal matrix?

Suppose $A$ is a real $n\times n$ normal matrix, and consider the matrix $$B=(I_n-\theta A)^{-1}(A+A^T)(I_n-\theta A^T)^{-1},$$ where $\theta$ is a real scalar such that $(I_n-\theta A)$ is ...
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### Transforming a tensor representing matrix to the eigen system

Say I have some mapping in 2D t(v) = ... that is a tensor. I can find a matrix 2x2 T that represent this tensor, and find the ...
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### Getting a feel for the transformation A on vector x which lies outside of any eigenspace

In one of his videos, after 13:25 Sal starts to talk about the interpretation of the eigenvectors and how they relate to a vector $x$ being transformed by the ...
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### diagonalization of a bisymmetric matrix

Is there some way to easily diagonalize a rank $n$ bisymmetric Toeplitz matrix with only zeros on its main diagonal? Direct calculation is out of the question, I need some trick... thanks! Addendum: ...
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I'm looking for a stable method to compute $A^n$, where $A$ is the following defective $12 \times 12$ matrix: $$A = \left(\begin{array}{cccc|cccc|cccc} \frac{1}{2} & \frac{1}{2} & 0 & 0 ... 1answer 40 views ### There are n! different unitary similarities that give L_i \sim L_j. Or are there more? Given a matrix L of dimension n \times n which is lower triangular, and with distinct elements along the diagonal, there is a way to unitarily transform it into a different lower triangular matrix ... 1answer 75 views ### Eigenvector of A to given Eigenvalue which requires row swapping to get reduced echelon form Given the Matrix$$A = \left(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -3 & -3 \end{matrix}\right)$$calculate the eigenvalues and the corresponding eigenvectors of ... 1answer 113 views ### Verifying Orthogonality of Eigenvectors How do you 'verify' the orthogonality of the eigenvectors of a matrix, let's say {\bf A} , for example? I came across the result that a matrix {\bf A} has orthogonal eigenvectors if {\bf ... 1answer 83 views ### Problem with null Eigen vectors. Please help me calculate the Eigen vectors of this matrix.$$\begin{pmatrix} 3 & 0 & 1\\ 1 & 3 & 0\\ 0 & 1 & 3 \end{pmatrix}$$The first vector comes out to be null, no clue ... 2answers 578 views ### Minimal polynomials and characteristic polynomials I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices. When are the minimal polynomial and characteristic polynomial the ... 1answer 408 views ### Upper Triangular Form of a Matrix I am trying to find the upper triangular form of B and an invertible matrix C such that B=C^{-1}AC where A is given by the following: A = \pmatrix{1&1\\ -1&3} The ... 1answer 74 views ### Find eigenvalues and eigenvectors: strange case Why is this expression:$$\begin{pmatrix} \frac{k+mg}{l} & -k\\ -k & \frac{k+mg}{l} \end{pmatrix} \begin{pmatrix} \rho_1\\\rho_2 \end{pmatrix}=\omega^2\begin{pmatrix} m & 0\\ 0 & m ...
I am studying Generalized Eigenvectors. It seems that we can define them as $\mathbf{p}_i$ in this equation: $$(\mathbf{A}-\lambda\mathbf{I})^{k}\mathbf{p}_i = \mathbf{0}$$ in which $k$ is the ...
On the final exam of my linear algebra class, the instructor asked us to compute the cube root of matrix $$A = \begin{pmatrix}3 &2 & 0 \\ -4 &-3 &0\\ 0 &0& 8 \end{pmatrix}$$ I ...