Is there a way to diagonalise a tridiagonal symmetric matrix?

[gtfitzpatrick]

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 \left|A\right| = a_{}11\left|M_{}11\right| - (a_{}1)^{}2\left|B\right|

where \left|M_{}11\right| is the minor of a_{}11

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...

 

[gtfitzpatrick]

A = <br /> \begin{pmatrix}a11 &amp; a12 &amp; 0 &amp; 0 &amp; ... \\ a21 &amp; a22 &amp; a23 &amp; 0 &amp; ... \\ 0 &amp; a32 &amp; a33 &amp; a34 &amp; 0 &amp; ...\\ 0 &amp; 0 &amp; a43 &amp; a44 &amp; a45 &amp; ... \end{pmatrix}<br /> <br />

B = <br /> \begin{pmatrix}a33 &amp; a34 &amp; 0 &amp; 0 &amp; ... \\ a43 &amp; a44 &amp; a45 &amp; 0 &amp; ... \\ 0 &amp; a54 &amp; a55 &amp; a56 &amp; 0 &amp; ...\\ 0 &amp; 0 &amp; a65 &amp; a66 &amp; a67 &amp; ... \end{pmatrix}<br /> <br />

B = <br /> \begin{pmatrix}a22 &amp; a23 &amp; 0 &amp; 0 &amp; ... \\ a32 &amp; a33 &amp; a34 &amp; 0 &amp; ... \\ 0 &amp; a43 &amp; a44 &amp; a45 &amp; 0 &amp; ...\\ 0 &amp; 0 &amp; a54 &amp; a55 &amp; a56 &amp; ... \end{pmatrix}<br /> <br />

a12^{}2 = a12 x a21 because its symetric

 

[Mark44]

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 M_11 of entry a_11 is the (n - 1) x (n - 1) matrix whose upper-left entry is a_2. 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

 

[Physicslad78]

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