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