# Tagged Questions

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### Finding eigenvectors for the largest eigenvalue vs one with the largest absolute value

If I want to solve a generalized eigenvalue problem such as: $$A x = \lambda x$$ The problem is to find eigenvectors corresponding to the largest eigenvalues (sometimes in an optimization problem ...
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Suppose $V=V_0\oplus V_1$ be a $Z_2$-graded semi-simple lie algebra and, $\xi\in V_1$. The maps $ad_\xi \circ ad_\xi :V_0\longrightarrow V_0$ and $ad_\xi \circ ad_\xi :V_1\longrightarrow V_1$ are ...
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### Diagonalization of a matrix with change of basis

I was trying to diagonalize a not really nice matrix doing first a change of basis but I noticed that the two characteristic polynomials I get are different. Original matrix and its characteristic ...
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### Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
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### Quadratic form in canonical form relation [on hold]

The homogeneous quadratic form can be written as a matrix. It is also written as a canonical form by using orthogonal transformation. Why we are going for canonical form and what is the relation ...
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### Positive definite [on hold]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
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### Positive definite matrix. [on hold]

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
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### Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: $$A x = \lambda B x,$$ with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
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### Eigen values of AB and BA

let A be a linear transformation from $R^n$ to $R^m$, and B be a linear transformation from $R^m$ to $R^n$, it's easy to show that AB and BA has same eigen-value(except $0$). But my question is how ...
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### Matrix Spectral Radius and Induced Matrix Norms

Let $A$ be a matrix, and $\rho(A)$ be its spectral radius, $\|A\|_p$ be an norm induced from vector $p$-norm. (1) When $\rho(A)=\|A\|_2$ or $\rho(A)=\|A\|_1$, does $\|A\|_1=\|A\|_2$? (2) If the ...
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### Infinite Dimensional Vector Space Equivalence of Positive Matrix

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
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### Can the matrix product $PA$ be skew-symmetric with $P=P^T>0$ and $A$ Hurwitz?

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ is a real symmetric positive definite matrix. What will be ...