Properties

Understanding matrix and vector properties is key to unlocking deeper insights in linear algebra. 🧠

Learning Objectives

Learning objectives of the Properties section.

• Norm
• Transpose
• Inverse
• Eigenvalues

Summary Table

Summary of the Properties section.

Concept	Description	Example
L1-norm	Measures total absolute difference between two vectors (Manhattan distance).	\[\| \mathbf{u} - \mathbf{v} \|_1 = \sum_{i=1}^n |u_i - v_i|\]
L2-norm	Measures straight-line distance between two vectors (Euclidean distance).	\[\| \mathbf{u} - \mathbf{v} \|_2 = \sqrt{ \sum_{i=1}^n (u_i - v_i)^2 }\]
Transpose	Flips a matrix over its diagonal, turning rows into columns.	\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad A^\top = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}\]
Inverse (2×2 Matrix)	Reverses a matrix transformation when the determinant is nonzero.	\[A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}, \quad \det(A) = ad - bc\]
Eigenvalues/Vectors	\(\mathbf{v}\) is an eigenvector of \(A\) if \(A \mathbf{v} = \lambda \mathbf{v}\).	\[A \mathbf{v} = \lambda \mathbf{v}\]
Characteristic Equation	Equation whose solutions are the eigenvalues of \(A\).	\[\det(A - \lambda I) = 0\]