# Eigenvalues of a tridiagonal matrix

• Hassan2
In summary, there are special algorithms for finding the eigenvalues of tridiagonal matrices that do not "fill in the zeros" and are more efficient in terms of time and storage compared to general matrices. However, some general algorithms like the QR eigenvalue algorithm can also preserve the tridiagonality if implemented correctly.
Hassan2
Some algorithms of finding the eigenvalues of symmetric matrices first transform the matrix to a tridiagonal matrix which is similar to the original matrix and then find the eigenvalues of the tridiagonal matrix. . Are there special algorithms for a tridiagonal matrix, or do the same algorithms for general matrixes converge faster for a tridiagonal one?

A general n by n matrix has $n^2$ entries. An n by n "tri-diagonal" matrix has only 3n- 2 entries (not counting the ones that you know are 0 and don't have to keep track of). There are numerical algorithms for working with tri-diagonal matrices that do NOT "fill in the zeros" so, yes, there is an enormous saving in both time and storage working with tri-diagonal matrices.

Thanks HalsofIvy,

Do you know any of such algorithms?
The general algorithms that I know, like QR eigenvalue algorithm ,ruin the sparsity of the matrix in the very first iteration. Or maybe I misunderstood it.

Added: There was a mistake in my code. The QR transformation does preserve the tridiagonality.

Thanks again.

Last edited:

## 1. What is a tridiagonal matrix?

A tridiagonal matrix is a special type of square matrix where all the elements outside the main diagonal, as well as the elements on the main diagonal, are equal to zero. This means that the matrix has non-zero values only in the main diagonal and the two diagonals directly above and below the main diagonal.

## 2. What are eigenvalues of a tridiagonal matrix?

Eigenvalues of a tridiagonal matrix are the values that, when multiplied by a certain vector, result in a scalar multiple of that vector. In other words, they are the special values that do not change the direction of the vector when multiplied by the matrix.

## 3. How do you find the eigenvalues of a tridiagonal matrix?

To find the eigenvalues of a tridiagonal matrix, you can use various methods such as the QR algorithm, the bisection method, or the divide-and-conquer method. These methods involve converting the matrix to a simpler form and then solving for the eigenvalues using various mathematical techniques.

## 4. Why are eigenvalues of a tridiagonal matrix important?

Eigenvalues of a tridiagonal matrix are important because they provide valuable information about the matrix, such as its spectral radius, condition number, and stability. They are also used in various applications such as solving differential equations, image processing, and data compression.

## 5. Can a tridiagonal matrix have complex eigenvalues?

Yes, a tridiagonal matrix can have complex eigenvalues. This means that the eigenvalues have both a real and imaginary component. In fact, for a tridiagonal matrix with complex entries, the eigenvalues are always complex. However, for a tridiagonal matrix with real entries, the eigenvalues can be either real or complex.

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