Hankel Matrix
- Entries are constant along each anti-diagonal (the same value on each anti-diagonal).
- Common in signal processing to model time-invariant systems; also used for solving linear systems and in combinatorics.
- Typical properties cited: symmetry about the center, diagonal dominance, and positive semi-definiteness.
Definition
Section titled “Definition”A Hankel matrix is a square matrix with constant values along the anti-diagonals, starting from the second diagonal. It is a special type of Toeplitz matrix, which has constant values along the diagonals. Hankel matrices are commonly used in signal processing and other areas of applied mathematics.
Explanation
Section titled “Explanation”Because entries on each anti-diagonal are equal, a Hankel matrix exhibits patterned structure that is exploited in applications. The source describes uses in modeling time-invariant behavior of systems (for example, filters or amplifiers), solving systems of linear equations, and combinatorial counting problems. The text lists several commonly noted properties: symmetry about the matrix center, diagonal dominance (the sum of absolute values on each diagonal being greater than or equal to sums on other diagonals), and positive semi-definiteness (non-negative eigenvalues). These properties are presented as reasons Hankel matrices are useful for inversion, signal processing, and other applied tasks.
Examples
Section titled “Examples”Example 1
Section titled “Example 1”Matrix:
| 1 | 2 | 3 |
|---|---|---|
| 4 | 5 | 6 |
| 7 | 8 | 9 |
Description from source:
- This matrix has constant values along the anti-diagonals, starting from the second diagonal.
- The first diagonal contains the elements 1, 5, and 9, which are all different.
- The second diagonal contains the elements 2, 5, and 8, which are all the same.
- Similarly, the third diagonal contains the elements 3, 6, and 9, which are also the same.
Example 2
Section titled “Example 2”Matrix:
| 1 | 0 | 0 |
|---|---|---|
| 1 | 1 | 0 |
| 1 | 1 | 1 |
Description from source:
- In this matrix, the constant values along the anti-diagonals are 1, 1, and 1.
- The first diagonal contains the elements 1, 1, and 1, which are all the same.
- The second diagonal contains the elements 0, 1, and 1, which are also the same.
- The third diagonal contains the elements 0, 0, and 1, which are also the same.
Use cases
Section titled “Use cases”- Modeling time-invariant behavior of systems in signal processing (e.g., filters or amplifiers).
- Solving systems of linear equations in linear algebra.
- Counting arrangements in combinatorics.
Notes or pitfalls
Section titled “Notes or pitfalls”- The source highlights symmetry about the center as a key property: anti-diagonals having constant values implies the matrix is symmetric about its center.
- Diagonal dominance is presented as a property that aids inversion and solving linear systems.
- The source states Hankel matrices are always positive semi-definite, meaning their eigenvalues are non-negative.
Related terms
Section titled “Related terms”- Toeplitz matrix