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Mathematics

Linear Algebra

Canonical Forms

Bilinear Forms

Complex Numbers in Linear Algebra

Determinants and Properties

Diagonalization

Jordan Canonical Form

Matrix Decompositions

Linear Algebra Formulas

Linear Algebra in Coding Theory

Matrix Inequalities

Matrix Norms

Quadratic Forms

Types of Matrices

Rank and Nullity

Spectral Theorem

Singular Value Decomposition

Systems of Linear Equations

Types of Matrices

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Zero Matrix

A matrix in which all elements are zero. Example:

\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

Identity Matrix

A square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. Example:

\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Sparse Matrix

A matrix in which most of the elements are zero. Example:

\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 2 \end{bmatrix}

Triangular Matrix

A matrix where all the entries above the main diagonal (upper triangular) or below the main diagonal (lower triangular) are zero. Example (Upper):

\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}

Example (Lower):

\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \end{bmatrix}

Symmetric Matrix

A square matrix that is equal to its transpose. Example:

\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}

Skew-Symmetric Matrix

A square matrix whose transpose equals its negative. Example:

\begin{bmatrix} 0 & -2 & -3 \\ 2 & 0 & -5 \\ 3 & 5 & 0 \end{bmatrix}

Diagonal Matrix

A matrix where all entries outside the main diagonal are zero. Example:

\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}

Toeplitz Matrix

A matrix in which each descending diagonal from left to right is constant. Example:

\begin{bmatrix} 1 & 2 & 3 \\ 4 & 1 & 2 \\ 5 & 4 & 1 \end{bmatrix}

Orthogonal Matrix

A square matrix with real entries whose columns and rows are orthogonal unit vectors. Example:

\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}

Skew-Hermitian Matrix

A complex square matrix whose conjugate transpose equals its negative. Example:

\begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}

Circulant Matrix

A square matrix where each row vector is rotated one element to the right relative to the preceding row vector. Example:

\begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{bmatrix}

Block Matrix

A partitioned matrix composed of smaller matrices called blocks. Example:

\begin{bmatrix} A & B \\ C & D \end{bmatrix}

where

A, B, C,

and

D

are all matrices themselves.

Hankel Matrix

A matrix in which each ascending diagonal from left to right is constant. Example:

\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix}

Hermitian Matrix

A complex square matrix that is equal to its own conjugate transpose. Example:

\begin{bmatrix} 2 & 1+i \\ 1-i & 3 \end{bmatrix}

Unitary Matrix

A square matrix with complex entries that when multiplied by its conjugate transpose results in the identity matrix. Example:

\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{i}{\sqrt{2}} \end{bmatrix}

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