Matrix

linear-algebra

Definition

Matrix

A $m\times n$ matrix $M$ with coefficients $a_{ij}\in F$, where $F$ is a field, is a rectangular array of coefficients:

$$M=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]$$

where $a_{ij}\in F$ with $i\in \{1,\dots ,m\}$ and $j\in \{1,\dots ,n\}$. The set of all $m\times n$ matrices with coefficients in $F$ is denoted by $F^{m\times n}$.

If $m=n$, the the matrix is called square matrix.

Properties

$$\begin{array}{rl}A\cdot I& =I\cdot A=A,\\ (A\cdot B)\cdot C& =A\cdot (B\cdot C),\\ (A+B)\cdot C& =A\cdot C+B\cdot C,\\ A\cdot (B+C)& =A\cdot B+A\cdot C,\\ (\lambda \cdot A)\cdot B& =A\cdot (\lambda \cdot B)=\lambda \cdot (A\cdot B),\\ (A+B{)}^T& =A^T+B^T,\\ (A\cdot B{)}^T& =B^T\cdot A^T,\\ (\lambda \cdot A{)}^T& =\lambda \cdot A^T.\end{array}$$

Arithmetic

Addition

Let $A,B\in F^{m\times n}$ be two matrices:

$$A+B:=\left[\begin{array}{cccc}{a}_{11}+b_{11}& a_{12}+b_{12}& \dots & a_{1n}+b_{1n}\\ a_{21}+b_{21}& a_{22}+b_{22}& \dots & a_{2n}+b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}+b_{m1}& a_{m2}+b_{m2}& \dots & a_{mn}+b_{mn}\end{array}\right]$$

Subtraction

Let $A,B\in F^{m\times n}$ be two matrices:

$$A-B:=\left[\begin{array}{cccc}{a}_{11}-b_{11}& a_{12}-b_{12}& \dots & a_{1n}-b_{1n}\\ a_{21}-b_{21}& a_{22}-b_{22}& \dots & a_{2n}-b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}-b_{m1}& a_{m2}-b_{m2}& \dots & a_{mn}-b_{mn}\end{array}\right]$$

Multiplication

Let $A=(a_{ij})\in F^{m\times n}$ and $B=(b_{jk})\in F^{n\times q}$ be two matrices:

$$\begin{array}{rl}{c}_{ik}& =a_{i1}{b}_{1k}+a_{i2}{b}_{2k}+\cdots +a_{in}{b}_{nk}\\ & =\sum _{j=1}^n{a}_{ij}{b}_{jk}\end{array}$$

Example:

$$\begin{array}{rl}A\cdot B& =\left[\begin{array}{ccc}1& 5& 2\\ 3& 2& 1\\ 0& 1& 2\end{array}\right]\cdot \left[\begin{array}{cc}1& 2\\ 5& 7\\ 0& 2\end{array}\right]\\ & =\left[\begin{array}{cc}1\cdot 1+5\cdot 5+2\cdot 0& 1\cdot 2+5\cdot 7+2\cdot 2\\ 3\cdot 1+2\cdot 5+1\cdot 0& 3\cdot 2+2\cdot 7+1\cdot 2\\ 0\cdot 1+1\cdot 5+2\cdot 0& 0\cdot 2+1\cdot 7+2\cdot 2\end{array}\right]\\ & =\left[\begin{array}{cc}26& 41\\ 13& 22\\ 5& 11\end{array}\right]\end{array}$$

Transpose

Definition

Matrix Transpose

The transpose of a matrix $M$ is the matrix with the rows and columns interchanged:

$$\begin{array}{rl}{M}^T& ={\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]}^T\\ & =\left[\begin{array}{cccc}{a}_{11}& a_{21}& \dots & a_{m1}\\ a_{12}& a_{22}& \dots & a_{m2}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& \dots & a_{mn}\end{array}\right]\end{array}$$

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Power

Raising a matrix $M$ to the power of $k$ means matrix-multiplying matrix $M$ $k$ times:

$$M^k=\underset{k\text{ times }}{\underbrace{M\times M\times \cdots \times M}}$$

Special Matrices

Square Matrix

Definition

Square Matrix

A square matrix $M$ is a matrix with equal number of rows and columns.

A square matrix is called symmetric matrix if:

$$M=M^T$$

where $M^T$ is the transposition of $M$.

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Symmetric Matrix

Definition

Symmetric Matrix

A matrix $M$ is called symmetric if the tranposed matrix $M^T$ is equal to $M$.

$$M=M^T$$

where $M^T$ is the transposition of $M$.

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Invertible Matrix

Definition

Invertible Matrix

A square matrix $A\in F^{m\times n}$ is called invertible if there exists a matrix $A^{-1}\in F^{m\times n}$ such that:

$$\exists A^{-1}\in F^{m\times n}:A\cdot A^{-1}=A^{-1}\cdot A=I_n$$

The matrix $A^{-1}$ is called the inverse of $A$. Not invertible matrices are called singular matrices.

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Rank

Rank

The rank of a matrix $M$ is the dimension (number of basis vectors) of the linear span of its column or row vectors.

To determine the rank of a matrix, the gaussian elimination algorithm can be used.