Matrix Types | Sahithyan's Notes

Diagonally Dominant Matrix

An $n \times n$ matrix $A$ is diagonally dominant iff

|a_{ii}| \geq \sum_{j=1, j \neq i}^{n} |a_{ij}|

holds for each $i = 1, 2, \dots, n$ .

Strictly Diagonally Dominant Matrix

$A$ is strictly diagonally dominant iff the inequality above is strict for each $i$ . Non-singular.

Gaussian elimination on $Ax = b$ then proceeds without row or column interchanges, and the computation is stable with respect to round-off error growth.

Positive Definite Matrix

See Positive Definite Matrix for the definition and eigenvalue characterization.

A leading principal submatrix $A_k$ of $A$ is the $k \times k$ block

A_k = \begin{bmatrix} a_{11} & \cdots & a_{1k} \\ \vdots & \ddots & \vdots \\ a_{k1} & \cdots & a_{kk} \end{bmatrix}

for some $1 \leq k \leq n$ .

A symmetric matrix $A$ is positive definite iff every leading principal submatrix has a positive determinant.

A symmetric matrix $A$ is positive definite iff Gaussian elimination on $Ax = b$ can be performed without row interchanges with all pivot elements positive. The computation is then stable with respect to round-off error growth.

A = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}

is positive definite: its leading principal submatrices have determinants $2$ , $3$ , and $4$ , all positive.

Convergent Matrix

An $n \times n$ matrix $A$ is convergent to $B$ iff

\lim_{k \to \infty} A^k = B

For a square matrix $A$ , the following are equivalent:

$A$ is convergent to $0$ , i.e. $\lim_{k \to \infty} A^k = 0$
$\lim_{k \to \infty} \lVert A^k \rVert_m = 0$ for some matrix norm $\lVert \cdot \rVert_m$
$\rho(A) < 1$