Matrix
In Linear Algebra a $m \times n$ matrix $A$ is a rectangular array with $m$ rows and $n$ columns:
$A = \begin{bmatrix} a_{11} & a_{12} & … & a_{1n}\ a_{21} & a_{22} & … & a_{2n}\ … & … & … & … \ a_{m1} & a_{m2} & … & a_{mn} \end{bmatrix}$
${a_{ij}}$ (or $(A)_{ij}$) are components of the matrix $A$
if $m = n$, then $A$ is called ‘‘rectangular’’
$(a_{11}, a_{22}, …, a_{nn})$ are diagonal elements
Operations
- Matrix Multiplication: Can multiply a matrix by a scalar, by a vector or by another matrix
- Matrix Transposition
- Inversion
- …
Types
Matrices can be:
- square $n \times n$ and rectangular $m \times n$
- Rank-1 Matrices
- Identity matrices
- Symmetric Matrices
- Orthogonal Matrices
- Rotation Matrices
- Similar Matrices
- Positive-Definite Matrices
Decompositions
- LU Decomposition: $A = LU$ where $L$ is lower triangular and $U$ is upper triangular
- QR Decomposition: $A = QR$ where $Q$
- Eigendecomposition: $A = S \Lambda S^{-1}$ with diagonal $\Lambda$
- special case of EVD: Spectral Theorem: $A = Q \Lambda Q^T$ with diagonal $\Lambda$ and Orthogonal $Q$
- Singular Value Decomposition: $A = U \Sigma V^T$ with diagonal $\Sigma$ and orthogonal $U$ and $V$
Matrices as Vectors
We can see matrices as vectors, and they also can form Vector Spaces
- see Matrix Vector Spaces
- they have inner product (element-wise) and norm (Frobenius Norm)
Sources
- Linear Algebra MIT 18.06 (OCW)
- Курош А.Г. Курс Высшей Алгебры