Is there a way to diagonalise a tridiagonal symmetric matrix?

In summary, the conversation discusses the properties of a symmetric and tridiagonal matrix A and its minor a_11. It is then shown that the determinant of A can be calculated using the minor and the determinant of a smaller matrix B, formed by deleting the first two rows and columns of A. There is also a question about diagonalizing a tridiagonal symmetric matrix using similarity transformations or block matrices.
  • #1
gtfitzpatrick
379
0
The matrix A is symmetric and tridiagonal.
If B is the matrix formed from A by deleting the first two rows and columns, show that [tex]\left|A\right|[/tex] = a[tex]_{}11[/tex][tex]\left|M_{}11\right|[/tex] - (a[tex]_{}1[/tex])[tex]^{}2[/tex][tex]\left|B\right|[/tex]

where [tex]\left|M_{}11\right|[/tex] is the minor of a[tex]_{}11[/tex]

I know what a symmetric tridiagonal matrix is.
Is the minor oa a11 not just a11, the minor is the deterninant of a smaller part of a matrix right? but since a11 in only one entry is it not the minor as well?

i'm not sure where to start this...
 
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  • #2
A = [tex]
\begin{pmatrix}a11 & a12 & 0 & 0 & ... \\ a21 & a22 & a23 & 0 & ... \\ 0 & a32 & a33 & a34 & 0 & ...\\ 0 & 0 & a43 & a44 & a45 & ... \end{pmatrix}

[/tex]

B = [tex]
\begin{pmatrix}a33 & a34 & 0 & 0 & ... \\ a43 & a44 & a45 & 0 & ... \\ 0 & a54 & a55 & a56 & 0 & ...\\ 0 & 0 & a65 & a66 & a67 & ... \end{pmatrix}

[/tex]

B = [tex]
\begin{pmatrix}a22 & a23 & 0 & 0 & ... \\ a32 & a33 & a34 & 0 & ... \\ 0 & a43 & a44 & a45 & 0 & ...\\ 0 & 0 & a54 & a55 & a56 & ... \end{pmatrix}

[/tex]

a12[tex]^{}2[/tex] = a12 x a21 because its symetric
 
  • #3
The minor of an entry in a matrix is the submatrix made up of all rows and columns that don't include that entry. For example, the minor [itex]M_11[/itex] of entry [itex]a_11[/itex] is the (n - 1) x (n - 1) matrix whose upper-left entry is [itex]a_2[/itex]. A minor is a matrix, and is different from its determinant.

You're on the right track. Matrix B is as you show it in the first equation for B, with its upper-left entry of a33. I don't know what the other equation for B represents with its upper-left entry of a22.

To evaluate |A| by minors, you'll get a11 * M11 - a12 * M12, where M12 is the submatrix of all entries not in row 1 and column 2. The 1st column of M12 has only one nonzero entry in it: a21 (which by symmetry = a12). When you expand A12, going down the first column, you'll get a21 * |B|. Be sure to include the appropriate signs associated with a12 and a21.

Hope that helps
 
  • #4
I have a question on tridiagonal symmetric matrices..Is there a way to diagonalise it analytically like applying similarity transformations or in terms of Block matrices?..Thanks
 

1. What is a symmetric tridiagonal matrix?

A symmetric tridiagonal matrix is a special type of square matrix that has non-zero elements only on the main diagonal, the diagonal above it, and the diagonal below it. This means that the matrix has a pattern of non-zero elements that form a "tridiagonal" shape.

2. What are the properties of a symmetric tridiagonal matrix?

A symmetric tridiagonal matrix has the following properties:

  • It is a square matrix.
  • It is symmetric, meaning that it is equal to its own transpose.
  • It is tridiagonal, meaning that it has non-zero elements only on the main diagonal, the diagonal above it, and the diagonal below it.
  • It is a banded matrix, meaning that it has a limited number of non-zero diagonals.

3. How is a symmetric tridiagonal matrix useful in scientific applications?

Symmetric tridiagonal matrices are useful in many scientific applications, such as solving systems of linear equations, numerical analysis, and differential equations. They are also commonly used in algorithms for solving eigenvalue problems.

4. How is a symmetric tridiagonal matrix different from a diagonal matrix?

A symmetric tridiagonal matrix is different from a diagonal matrix in that it has non-zero elements on the diagonal above and below the main diagonal, whereas a diagonal matrix only has non-zero elements on the main diagonal. This means that a symmetric tridiagonal matrix has a more structured pattern of non-zero elements compared to a diagonal matrix.

5. How can a symmetric tridiagonal matrix be created?

A symmetric tridiagonal matrix can be created in several ways, such as using a specific formula or algorithm, or by manipulating a given matrix to have the necessary properties. For example, a symmetric tridiagonal matrix can be created by using the tridiagonal matrix algorithm or by extracting the necessary elements from a larger matrix. It can also be created using specific functions or commands in programming languages like MATLAB or Python.

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