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An eigenvalue $\lambda$ of a matrix $A$ is a scalar such that there exists a non-zero vector $v$ where $A\mathbf{v} = \lambda\mathbf{v}$. Example: For $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$, one eigenvalue is $\lambda = 5$ because $(A - 5I)\mathbf{v} = 0$ has a non-trivial solution.

The trace of a square matrix $A$, denoted as $tr(A)$, is the sum of its diagonal elements. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $tr(A) = 1 + 4 = 5$.

An eigenvector of a matrix $A$ is a non-zero vector $\mathbf{v}$ such that $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$, the eigenvalue. Example: For $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$ and $\lambda = 5$, an eigenvector is $\mathbf{v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ because $A\mathbf{v} = 5\mathbf{v}$.

LU decomposition expresses a matrix $A$ as the product of a lower triangular matrix $L$ and an upper triangular matrix $U$. Example: For $A = \begin{bmatrix} 4 & 3 \\ 6 & 3 \end{bmatrix}$, $L = \begin{bmatrix} 1 & 0 \\ 1.5 & 1 \end{bmatrix}, U = \begin{bmatrix} 4 & 3 \\ 0 & -1.5 \end{bmatrix}$, so $A = LU$.

The dot product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is a scalar $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + ... + a_nb_n$. Example: For $\mathbf{a} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, $\mathbf{b} = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$, $\mathbf{a} \cdot \mathbf{b} = 1\cdot4 + 2\cdot5 + 3\cdot6 = 32$.

The column space of a matrix $A$, $col(A)$, is the span of its column vectors. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $col(A)$ is the plane spanned by $\begin{bmatrix} 1 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \end{bmatrix}$ in $\mathbb{R}^2$.

The row space of a matrix $A$, denoted as $row(A)$, is the subspace generated by its row vectors. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $row(A)$ is the plane spanned by $\begin{bmatrix} 1 & 2 \end{bmatrix}$ and $\begin{bmatrix} 3 & 4 \end{bmatrix}$ in $\mathbb{R}^2$.

The cross product of vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^3$ is a vector $\mathbf{c} = \mathbf{a} \times \mathbf{b}$, calculated as $\mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$. Example: For $\mathbf{a} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $\mathbf{b} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$, $\mathbf{a} \times \mathbf{b} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$.

Given a scalar $k$ and a matrix $A$, the product $kA$ is obtained by multiplying each element of $A$ by $k$. Example: For $k = 3$ and $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $3A = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix}$.

The product of two matrices $A$ and $B$, is a new matrix $C$ where each element $c_{ij}$ is the dot product of the ith row of $A$ and the jth column of $B$. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, $AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$.

The kernel of a matrix $A$, denoted as $ker(A)$, is the set of vectors $\mathbf{v}$ such that $A\mathbf{v} = \mathbf{0}$. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$, $ker(A) = \left\{ t\begin{bmatrix} -2 \\ 1 \end{bmatrix} | t \in \mathbb{R} \right\}$.

For a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $|A| = ad - bc$. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $|A| = 1\cdot4 - 2\cdot3 = -2$.

The rank of a matrix $A$ is the maximum number of linearly independent column vectors (or row vectors) in the matrix. Example: For $A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$, since the third row is a linear combination of the first two, $rank(A) = 2$.

The identity matrix $I_n$ of size $n$ is a square matrix with ones on the main diagonal and zeros elsewhere. Example: $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

A diagonal matrix $D$ is a matrix where all off-diagonal elements are zero. Example: $D = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$.

For a square matrix $A$, the inverse is $A^{-1} = \frac{1}{|A|} \cdot adj(A)$ where $|A|$ is the determinant and $adj(A)$ is the adjugate. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}$.

For two matrices $A$ and $B$ of the same dimension, their sum $C = A + B$ is found by adding corresponding elements. Example: For $A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}$, $A + B = \begin{bmatrix} 6 & 10 \\ 8 & 12 \end{bmatrix}$.

The norm of a matrix $A$, denoted $||A||$, gives a measure of its size. Common norm is the Frobenius norm $||A||_F = \sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^2}$. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $||A||_F = \sqrt{1^2 + 2^2 + 3^2 + 4^2} = \sqrt{30}$.

For a system of linear equations $AX = B$ with a square matrix $A$, the solution for each variable $x_i$ is given by $x_i = \frac{|A_i(B)|}{|A|}$, where $A_i(B)$ is matrix $A$ with its ith column replaced by vector $B$. Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 \\ 6 \end{bmatrix}$, $x_1 = \frac{|\begin{bmatrix} 5 & 2 \\ 6 & 4 \end{bmatrix}|}{|\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}|} = \frac{8}{-2} = -4$.