# Tagged Questions

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### Given a matrix $A$, find $A^n$

Given the matrix $$A = \left[{9\atop20}{-4\atop-9}\right]$$ how do I find $A^7$ or $A^{54}$ or $A^{2008}$ (etc.) ? I know I need the eigenvalues of A, but I'm not sure what to do afterwards. Is the ...
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### Fundamental matrix for a given system of equation

Question is to find the fundamental matrix(F(t)) satisfying F(0)=I for the given system of equation below. $$x' =\left(\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right)x$$ My solution ...
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### Find The Eigenvalues and Eigenvectors of the Hermitian Matrix

Find the eigenvalues and eigenvectors of the $2\times2$ hermitian matrix. $$\pmatrix{\epsilon_1&|V|e^{i\alpha}\\ |V|e^{-i\alpha}&\epsilon_2}$$ I know to find eigenvalues, you use ...
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### Complex Matrix Limit

If $A$ is an $n \times n$ complex matrix, show that if $\lim_{k\rightarrow\infty}||A^kv||=0$ for every vector $v \in \Bbb C^n$, then $|\lambda|\leq1$ for every eigenvalue $\lambda$ of $A$.
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### 2x2 matrix and convergence under iteration connected to eigenvectors and eigenvalues. [duplicate]

When working with a general matrix $A=$ \begin{bmatrix}a & b\\c & d\end{bmatrix}, I find that the eigenvalues are: $\lambda = \frac{d+a \pm \sqrt{(d-a)^2+4bc}}{2}$ I then find that ...
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### Coordinate System Rotation and Cross Term

If I have a conic equation $$5x^2 - 4xy + 8y^2 = 36$$ and $\left[\begin{array}{cc} 5 & -2\\ -2 & 8 \end{array}\right]$ in matrix form, whose eigenvalues are 4 and 9, how would I rotate ...
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### Finding Eigensystem of Hermitian Matrix

I am trying to find the system of eigenvalues and corresponding normalized eigenvectors for the following Hermitian matrix: $$\mathbf{H}=\begin{pmatrix}10 & 3i \\ -3i & 2\end{pmatrix}$$ I ...
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### Hermitian matrices, non-zero

If I have a hermitian matrix whose eigen values are non negative, and the trace=0, must the matrix=0? I gather that the eigen values must all be 0, but I could not find an example of a hermitian ...
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### Eigenvalues of sum of Hermitian matrices

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### Eigenvectors 2x2

I have matrix $\begin{bmatrix} 1&0 \\ 4&1 \end{bmatrix}$ with repeated eigenvalue of 1. How do I find the eigenvectors? All I seem to get is $\begin{bmatrix} 0\\ 0 \end{bmatrix}$
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### B is A's Jordan form, so that there's an invertible matrix P so that $A=PBP^{-1}$. How do I find P?

B is A's Jordan form, so that there's an invertible matrix P so that $A=PBP^{-1}$. How do I find P? I tried solving, and here's a detailed path of my solution. Any help is very much appreciated! For ...
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### Prove the operators $T+U$ and $U$ have the same eigenvalues where $T$ is nilpotent

Let $V$ be an $n$-dimensional vector space on $\mathbb{C}$, and $T$ a nilpotent operator on $V$. Let $U$ be in $L(V)$ s.t. $UT = TU$. Prove that the operators $T+U$ and $U$ have the same eigenvalues. ...
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### Find formulas for the entires of $M^n$ ($n\ge0$) (Eigen Vector/Values)

I need a little help with an eigen vector question, The question Let $M =\left|\begin{matrix} 1 & 1 \\ -36 & 13 \end{matrix}\right|$ Find formulas for the entries ...
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### Linear transformations and eigenvalues [duplicate]

Let $T: \mathbb C^n \rightarrow \mathbb C^n$ be linear. Let $\beta$ and $\gamma$ be any two ordered bases. Prove that the eigenvalues of $[T]_\beta$ and $[T]_\gamma$ are the same. Can anyone provide ...
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### Proof for diagonalizable matrix

Let $A \in M_n(\mathbb C)$ be invertible. Prove that $A$ is diagonalizable if and only if $A^{-1}$ is diagonalizable. This is what I have for one direction of the proof: Suppose $A$ is ...
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### Proving that a matrix has positive eigenvalues.

Let's say that $\mathrm{A}$ is a diagonalizable matrix with distinct nonzero eigenvalues. Prove that $\mathrm{A}^2$ has positive eigenvalues. Ok I have a general idea how to do this, but not sure ...
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### $A= SJS^{-1}$ Where $J$ is in Jordan Normal Form with One Eigenvalue

If we have that $A= SJS^{-1}$ where $J$ is in Jordan Normal Form with one eigenvalue, and we subtract $\lambda$ from $A$, why can we say that $(SJS^{-1}-\lambda I)^n=(S(J-\lambda I)S^{-1})^n$?
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### Assigning eigenvectors of a covariance matrix to the variables it was generated from

Preface: This is the follow-up question according to the insights I got from MvG on my earlier post. Think of an ellipsoid in the n-dimensional space defined by $$ell: (x-\mu)'A(x-\mu)=1.$$ Then one ...
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### Lanczos method to find the *largest* eigenpair

My question is whether the largest here means the eigenpair with the largest value or the largest absolute value?
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### Finding the eigen vectors of a 3x3 matrix

We are given the matrix: $$A = \begin{bmatrix}1 & 2 & 0\\2 & 1 & 0\\0 & 0 & 1\\\end{bmatrix}$$ I want to find the eigenvalues I did so by solving $$|\lambda I - A| = 0$$ ...
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### Information about $T$ if $T^2=0$

What can we say about $T$ if $T^2=0$? How can we prove that it has zero as an eigenvalue?
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### Linear Algebra: Orthonormal matrix and diagonal matrix

For (a), is the answer 2? For (b), are the eigenvectors [-1 0 1]T, [1 1 0]T and [1 -1 1]T and the corresponding eigenvalues 2 and 5? Also, Can someone explains why I can't obtain the eigenvector ...
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### Eigenvectors are linearly independent?

Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent. Could someone give me a geometric interpretation of the theorem? Thanks!
Let $B$ be an invertible $n\times n$ complex matrix. Prove that there exist $n\times n$ complex matrices $A, C$ such that the following three conditions are satisfied simultaneously: (i) $B = AC$ ...
If $\boldsymbol{A}$ is a rank deficient Hermitian matrix are the following true? 1) Is $<\boldsymbol{x}_j,\boldsymbol{x_k}>=0$ when $\lambda_j=\lambda_k$? 2) Is ...