3D Matrix Visualizer

Symmetric Matrix

A Symmetric Matrix is a matrix that is mirrored across the diagonal. Formally, a matrix is symmetric if it equals its transpose: A = A^T

Skew-Symmetric Matrix

A Skew-Symmetric Matrix is a matrix where the lower-triangle is the sign-flipped version of the upper-triangle. Formally: A = -A^T

Diagonal Matrix

A Diagonal Matrix is a matrix that has all zeros on the off-diagonals, and only the diagonal elements (going from top-left to bottom-right) may contain non-zero elements. Formally: \[ a_{i,j} = \begin{cases} \alpha_n & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} \]

Triangular Matrix

A Triangular Matrix is a matrix that can come in 2 forms:

upper-triangular: only the diagonal and the elements above it may contain non-zero elements
lower-triangular: only the diagonal and the elements below it may contain non-zero elements

Orthogonal Matrix

An Orthogonal Matrix is a matrix that satisfies the following two criteria:

All of its columns are pairwise orthogonal. It means that the dot product between any two columns is 0.
Each column is unit magnitude. Formally: \( \| Q_i \| = 1 \)

Formally: \[ \langle Q_i, Q_j \rangle = \begin{cases} 1, & \text{if } i = j \\ 0, & \text{if } i \neq j \end{cases} \]

Toeplitz Matrix

A Toeplitz Matrix is a matrix in which each descending diagonal contains the same element. Formally: A_i,j = A_i+1,j+1 = a_i-j

Hankel Matrix

A Hankel Matrix is a square matrix in which each ascending diagonal contains the same element