Questions tagged [eigenvalues-eigenvectors]

Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.

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Derivative of eigenvalue of matrix with respect to its elements

Assuming that matrix $A$ is positive semidefinite and that $\lambda$ denotes the eigenvalue, I would like to compute the following gradient $$\nabla_A \lambda(A)$$ I wanted to set this problem up ...
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Continuous dependence of eigenvalues on tangent-space endomorphisms

Consider the following setting: Let $M$ be a smooth manifold and suppose that $T \colon M \to T^{(1,1)}(M)$ is a smooth section, that is, $T_x \in \mathrm{End}(T_xM)$ for every $x \in M$. Moreover, ...
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How to find the diagonal matrix when transformation and conversion matrices are known?

I am stuck with finding the diagonal matrix for eigenvalues. Given the matrix T = \begin{bmatrix}6&-1\\2&3\end{bmatrix} and change of basis matrix C= (whose columns are eigenvectors of T) \...
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How to get positional coordinates from distance matrix? (follow-up on previous question)

I have a distance matrix D_ij. I would like to compute the positional coordinates that yield this given distance matrix. I browsed the stack exchange forums and ...
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eigenvalue of a giant matrix.can't see the pattern to solve [on hold]

\begin{bmatrix} -1-s&a&b&c&0&d \\a&-s&b&c&d&0\\b&c&-s&d&0&a\\c&d&0&-s&a&b\\d&0&a&b&-s&c\\0&a&...
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Finding eigenvalues without doing calculations

Find, without doing any calculations, the eigenvalues of the following linear transformations from $\mathbb{R}^2 \to \mathbb{R}^2$: $A)\quad$ Projection on a straight line that contains $(0,0)$ ...
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All eigenvectors of a real symmetric matrix are orthogonal to one another.

I did part of the proof where the eigenvectors corresponds to distinct eignevalues, where at the conclusion we get (λi - λj)xi'xj = 0 which follows that xi'xj = 0, where λi ≠ λj. I want to know how ...
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