Question

A matrix $A$ is said to have bandwidth $k$ if all entries that are more than $k$ slots away from the main diagonal are zero: $a_{i j}=0$ whenever $|i-j|>k$. (a) Show that a tridiagonal matrix has band width 1 . (b) Write down an example of a $6 \times 6$ matrix of band width 2 and one of band width 3. (c) Prove that the $L$ and $U$ factors of a regular banded matrix have the same band width. (d) Find the $L U$ factorization of the matrices you wrote down in part (b). (e) Use the factorization to solve the system $A \mathbf{x}=\mathbf{b}$, where $\mathbf{b}$ is the column vector with all entries equal to 1 . (f) How many arithmetic operations are needed to solve $A \mathbf{x}=\mathbf{b}$ if $A$ is banded? ( $g$ ) Prove or give a counterexample: the inverse of a banded matrix is banded.

    A matrix $A$ is said to have bandwidth $k$ if all entries that are more than $k$ slots away from the main diagonal are zero: $a_{i j}=0$ whenever $|i-j|>k$. (a) Show that a tridiagonal matrix has band width 1 . (b) Write down an example of a $6 \times 6$ matrix of band width 2 and one of band width 3. (c) Prove that the $L$ and $U$ factors of a regular banded matrix have the same band width. (d) Find the $L U$ factorization of the matrices you wrote down in part (b). (e) Use the factorization to solve the system $A \mathbf{x}=\mathbf{b}$, where $\mathbf{b}$ is the column vector with all entries equal to 1 . (f) How many arithmetic operations are needed to solve $A \mathbf{x}=\mathbf{b}$ if $A$ is banded? ( $g$ ) Prove or give a counterexample: the inverse of a banded matrix is banded.

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 1, Problem 15

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Step 1

- A tridiagonal matrix is a matrix where only the main diagonal, the diagonal above the main diagonal, and the diagonal below the main diagonal have non-zero entries. This means that for a tridiagonal matrix $A$, $a_{ij} = 0$ whenever $|i-j| > 1$. Thus, by Show more…

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A matrix $A$ is said to have bandwidth $k$ if all entries that are more than $k$ slots away from the main diagonal are zero: $a_{i j}=0$ whenever $|i-j|>k$. (a) Show that a tridiagonal matrix has band width 1 . (b) Write down an example of a $6 \times 6$ matrix of band width 2 and one of band width 3. (c) Prove that the $L$ and $U$ factors of a regular banded matrix have the same band width. (d) Find the $L U$ factorization of the matrices you wrote down in part (b). (e) Use the factorization to solve the system $A \mathbf{x}=\mathbf{b}$, where $\mathbf{b}$ is the column vector with all entries equal to 1 . (f) How many arithmetic operations are needed to solve $A \mathbf{x}=\mathbf{b}$ if $A$ is banded? ( $g$ ) Prove or give a counterexample: the inverse of a banded matrix is banded.

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