What is Tridiagonalization?

What is Tridiagonalization?

If A is a real symmetric n×n matrix, then A is orthogonally similar to a real tridiagonal n×n matrix T, i.e. there exists an orthogonal matrix U such that UT A U = T. A Householder transformation is an orthogonal transformation of the form.

What is the dimension of a tridiagonal matrix?

A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. In particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q/2 = n — the dimension of the tridiagonal.

What is symmetric tridiagonal matrix?

A tridiagonal matrix A is also symmetric if and only if its nonzero elements are found only on the diagonal, subdiagonal, and superdiagonal of the matrix, and its subdiagonal elements and superdiagonal elements are equal; that is: (aij = 0 if |i-j| > 1) and (aij = aji if |i-j| = 1)

How do you solve a tridiagonal matrix?

The system can be efficiently solved by setting Ux = ρ and then solving first Lρ = r for ρ and then Ux = ρ for x. The Thomas algorithm consists of two steps. In Step 1 decomposing the matrix into M = LU and solving Lρ = r are accomplished in a single downwards sweep, taking us straight from Mx = r to Ux = ρ.

What is tridiagonal matrix write an example?

A tridiagonal matrix is a matrix that has non-zero elements only at the main diagonal, diagonal below and above it. All other elements are zero. For this reason tridiagonal matrices of dimension smaller than or equal to 3 seem meaningless. Example 1: [a11, a22, 0 , 0 , 0 , 0 ]

How do you solve Thomas’s algorithm?

The Thomas algorithm consists of two steps. In Step 1 decomposing the matrix into M = LU and solving Lρ = r are accomplished in a single downwards sweep, taking us straight from Mx = r to Ux = ρ. In step 2 the equation Ux = ρ is solved for x in an upwards sweep.

Is Thomas algorithm an iterative method?

Explanation: Thomas algorithm solves a system of equations with non-repeated sequence of operations. It is a direct method to solve the system without involving repeated iterations and converging solutions.

How do you find the determinant of a tridiagonal matrix?

Let be an -by- bordered tridiagonal matrix. Let and be vectors of the form (2.2) and B − 1 h , respectively. Then the determinant of is given by det ( A ) = ( ( d n − δ ) ⋅ ∏ i = 1 n − 1 c i ) | λ = 0 , where δ = ∑ i = 1 n − 1 p i g i .

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