# Questions tagged [tridiagonal-matrices]

Relating to all $n\times n$ matrices $(A)$ with the property $a_{i,j}=0$ if $|j-i|>1$

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### Instability of cubic splines

I'm supposed to show, that the calculation of cubic splines with tridiagonal matrices is unstable. To show this I'm supposed to consider the function $s_1: [x_0, x_n] \to \mathbb{R}$ which is a spline ...
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### linear transformation, is diagonalizable? all values of a?

People i have the next problem. Let $f: \mathbb R_3 → \mathbb R_3$ be the linear transformation such that https://i.stack.imgur.com/qLTqd.png with $\mathcal B = \{(1,0, −1), (0,0,1), (0,1 , 0)\}$. The ...
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### Finding the eigenvalues of a discrete laplacian on an infinite lattice

If we define the Laplacian as a square matrix with zeroes on the diagonal, and -1 on the diagonals exactly above and below the main diagonal, and 0 everywhere else, how would one go about finding its ...
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### Show that $Q$ is Hessenberg for QR Factorization of tridiagonal matrices

Let $T$ be a tridiagonal, symmetric matrix in $\mathbb{R}^{n\times n}$. The QR algorithm of $T$ with shifts is defined is as follows; $$T^{(k)} − \mu I = Q^{(k)}R^{(k)}$$ (where right-hand side is a ...
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### How to show that two Hermitian tridiagonal matrices are similar?

Given two tridiagonal hermitian matrices A,B with $a_i\in \mathbb{R}$ and $b_i\in \mathbb{C}$ as follows \begin{align} A= \begin{pmatrix} a_{1} & |b_1| & \cdots & 0 \\ |b_1| & a_{...
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### Optimizing the inverse of a symmetric matrix with 3 non consecutive diagonal terms

Let $A$ be a $200x200$ matrix that is symmetric and has 3 diagonal terms. It is like a tridiagonal matrix but the diagonals are non-consecutive. Or like a banded matrix with bandwidth 100 but with ...
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### denseness of tridiagonal positive semidefinitness in tridiagonal positive definiteness

I know that pd (positive definite) matrices are dense in psd matrices. Let $A$ be TPSD (tridiagonal positive semi definite) with all entries positive. Does there exist a sequence of TPD matrices with ...
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