# Tagged Questions

Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is and eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective.

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### What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
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### Eigenvalue problem: Prove that all of the eigenvalues of $A$ are $1$.

Here's a cute problem that was frequently given by the late Herbert Wilf during his talks. Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. ...
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### What do eigenvalues have to do with pictures?

I am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article : Can someone explain this to me ?
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### How to diagonalize this matrix?

Consider the $n\times m$ matrix $M=[M_1, \ldots, M_m]$ where the $i$-th column reads $$M_i= \,^t(\underbrace{1,\ldots,1}_{a_i},0,\ldots,0)$$ where the $a_i$'s are given positive natural numbers. ...
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### Similar matrices have the same eigenvalues with the same geometric multiplicity

Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities. Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices ...
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### Find eigenvalues of a projection and explain what they mean

Suppose B represents the matrix of orthogonal (perpendicular) projection of $\mathbb{R}^{3}$ onto the plane $x_{2} = x_{1}$. Compute the eigenvalues and eigenvectors of B and explain their geometric ...
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### Why are all nonzero eigenvalues of the skew-symmetric matrices pure imaginary?

Assume that $A$ is an $n\times n$ skew-symmetric real matrix, i.e. $$A^T=-A.$$ Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and ...
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### Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks

Let $A$ be a block upper triangular matrix: $A = \left( \begin{matrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{matrix} \right)$ where $A_{1,1} ∈ C^{p×p}$, $A_{2,2} ∈ C^{(n-p)×(n-p)}$ Show ...
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### why do successive eigenfunctions have more oscillation?

I was told the following argument as to why successive eigenfunctions tend to have more oscillations: Suppose (without worrying about why) that the first eigenfunction has the least oscillation. ...
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### Eigenvector and its corresponding eigenvalue

For the following square matrix: $$\left( \begin{array}{ccc} 3 & 0 & 1 \\ -4 & 1 & 2 \\ -6 & 0 & -2 \end{array} \right)$$ Decide which, if any, of the ...
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### Determine whether a $3\times3$ matrix has a positive eigenvalue?

Given a $3\times3$ matrix is there a criterion capable of telling whether the matrix has a positive eigenvalue?
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### Eigenvalues of an interesting real symmetric matrix

Problem: Let A be a square matrix with all diagonal entries equal to 2, all entries directly above or below the main diagonal equal to 1, and all other entries equal to 0. Show that every eigenvalue ...
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### Eigen Value of a Linear Map

Let $V$ be the vector space of all continuous functions from $\mathbb{R}$ into $\mathbb{R}$ and let $T\colon V \rightarrow V$ be a linear map defined by $T(f)(x)=\int^{x}_{0}f(t)dt$. How can we prove ...
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### Finding the number of symmetric,positive definite $10 \times 10$ matrices having…

I was looking at old exam papers and I was stuck with the following problem: What is the number of symmetric,positive definite $10 \times 10$ matrices having trace equal to $10$ and determinant ...
I have a matrix $A$ and I introduce $(I+A)^{m},$ where $I$ is the identity matrix of same order with $A$ and $m$ is a positive integer. I want to show that if $(1+ \lambda )^m$ is a simple eigenvalue ...