12
votes
Matrix elements of exponential of tridiagonal matrices
Yes! Most methods to compute exponentials of large sparse matrices are based on computing directly $\exp(A)b$ for a given vector $b$ rather than the full matrix $\exp(A)$. Just take $b$ as a vector of ...
- 20.2k
8
votes
How can I calculate eigenvalues of a tridiagonal matrix?
Yes. For general tridiagonal matrices, see The Numerical Recipes, Chapter 11, or Golub-Van Loan. For symmetric tridiagonal matrices, you can do better, see Coakley/Rochlin's paper.
Coakley, Ed S.; ...
- 96.4k
7
votes
Accepted
Upper Bounds on the Largest Eigenvalue of Jacobi Matrices
The entries are nonnegative, so the dominant eigenvector has
all entries positive, and its eigenvalue is an increasing function of the $a_i$.
If each $a_i = 1$ then that eigenvalue is $1 + 2 \cos\frac\...
- 79.9k
3
votes
Accepted
Diagonalize almost symmetric tridiagonal matrix
Start with the matrix
$$
M' =
\begin{bmatrix} 0 & 0 \\
0 & a_2 & b_2 \\
& & \ddots \\
& & b_{n-2} & a_{n-1} & 0 \\
& & & 0 & 0
\end{bmatrix},
$$
i.e....
- 1,538
3
votes
Accepted
Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix
Defining the $2 \times 2$ transfer matrix
\begin{align}\tag{1}
Q = \begin{pmatrix} -\lambda & 1 \\ -1 & 0 \end{pmatrix},
\end{align}
the characteristic polynomial (CP) of the $M \times M$ ...
- 3,671
2
votes
Eigenvalues of symmetric tridiagonal matrices
The eigenvalues of a tridiagonal matrix are bounded by the maximum and minimum roots of a sequence of functions that form a chain sequence.
Specifically, given a general tridiagonal matrix
$$
A_n=
\...
- 21
2
votes
Is tridiagonal reduction the current best practice to compute eigenvalues of random matrices from the Gaussian ensembles (GOE, GUE, GSE)?
Q: What do people do in practice ?
The "practice" will be field specific, but in most of the physics applications I am aware of one needs not only the eigenvalues but also the eigenvectors. For ...
- 188k
2
votes
Accepted
Using permutation matrix to convert a matrix into tridiagonal matrix
The permutation matrix $P$ corresponding to the permutation
$$ \sigma \colon \begin{cases} i \mapsto 2i & \text{ if } i \in [1,n] \\ i \mapsto 2i - 2n - 1 & \text{ if } i \in [n+1,2n] \end{...
- 6,614
1
vote
Derivative of eigenvalues of a symmetric tridiagonal matrix built via the Lanczos-Arnoldi scheme
The magic of "not having to differentiate the eigenvectors" is known as the Hellmann–Feynman theorem. Let me walk you through it, at the level of a single eigenvalue, to answer the question ...
- 188k
1
vote
Sufficient conditions for invertibility of a block tridiagonal matrix
The following is a list of answers I know for some specific cases. However, they are not strong enough for my uses.
Simple conditions
A sufficient, but weak condition is that $C_i = 0$ for each $i$.
...
- 397
1
vote
Is tridiagonal reduction the current best practice to compute eigenvalues of random matrices from the Gaussian ensembles (GOE, GUE, GSE)?
Best practice for what please?: speed, accuracy, numerical stability, memory load, scaling on hardware parallel architectures...?
Do you need all eigenvalues or only some of them? Do you need their ...
- 1,026
1
vote
Action of square root of tridiagonal matrix product on vector
One can use Krylov subspace based methods, i.e. rational Krylov methods work well. There is a paper and matlab code that works out of the box: http://guettel.com/markovfunmv/.
The approach is a black-...
- 213
1
vote
Relation between the algebraic multiplicity of an eigenvalue and the subdiagonal elements of a symmetric tridiagonal matrix
The most natural explanation for this (in my view) lies in the fact that the eigenvectors of such a matrix solve a difference equation. More precisely, if we write $T_{nn}=b_n$, $T_{n,n+1}=T_{n+1,n}=...
- 24.2k
1
vote
Relationship between eigenvalues of summation of two matrices one is diagonal
This follows from the Interlacing Eigenvalue Theorem (Golub and Van Loan "Matrix Computations", 4th edition, Theorem 8.1.8), and holds for any real symmetric n by n $A$, whether or not Metzler or ...
- 1,517
1
vote
Accepted
Eigenvalues and eigenvectors of tridiagonal matrices
It seems that your question has already been answered here:
Eigenvalues of Symmetric Tridiagonal Matrices
No results for general tridiagonal matrices.
- 1,026
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