Vectors and Matrices — Study Notes and Flashcards Summary & Study Notes

📐 Overview

These notes summarize core concepts for vectors and matrices used in numerical computing and linear algebra. Focus is on how to create, index, and operate on vectors/matrices, plus key matrix properties and special matrix constructors. Mathematical symbols are shown using $...$ notation.

🧭 Creating Row and Column Vectors

A row vector is a 1×n array and a column vector is an n×1 array. For example, a row vector might be written as $[1;2;3]$ and a column vector as $\begin{bmatrix}1\2\3\end{bmatrix}$ (conceptually). In code, the difference is orientation: a row has one row and many columns, while a column has one column and many rows.

🔢 Indexing and Slicing

Indexing accesses individual elements using integer indices (often 1-based in math, 0-based in many programming languages). Slicing selects subranges of elements: e.g., selecting a subvector or a submatrix. Use colon notation conceptually: selecting rows $i$ to $j$ and columns $k$ to $l$ gives the submatrix $A(i!:!j,;k!:!l)$. Negative or logical indices may be supported in specific languages for reverse selection or masking.

🧱 Matrix Creation Methods

Common constructors:

• Bracket notation: direct listing such as $[1;2;;3]$ or $\begin{bmatrix}1 & 2\3 & 4\end{bmatrix}$ conceptually.
• linspace: creates linearly spaced vectors between two endpoints. Example: $\text{linspace}(a,b,n)$ produces $n$ values from $a$ to $b$.
• logspace: creates logarithmically spaced values. Example: $\text{logspace}(a,b,n)$ produces $n$ points between $10^a$ and $10^b$.

These methods are used to generate vectors for plotting, sampling, or building structured matrices.

➕➖✖ Matrix Operations

• Addition/Subtraction: Two matrices can be added or subtracted only if they have the same shape. Elementwise: $(A+B)${ij}=A{ij}+B_{ij}.
• Multiplication: Matrix multiplication $C=AB$ is defined when $A$ is $m\times p$ and $B$ is $p\times n$, producing an $m\times n$ matrix with $C_{ij}=\sum_k A_{ik}B_{kj}$. This is not elementwise by default.
• Elementwise multiplication (Hadamard) is a separate operation where elements are multiplied pairwise and requires same shape.
• Transpose: The transpose of $A$ is written $A^T$ and swaps rows and columns: $(A^T)${ij}=A{ji}. Useful for converting row vectors to column vectors and vice versa.

🔁 Inverse, Determinant, Rank, and Trace

• Inverse ($A^{-1}$): For square matrix $A$, the inverse satisfies $AA^{-1}=I$ when $A$ is nonsingular. Not all matrices have inverses. Numerically use stable methods (LU, QR) rather than naive inversion.
• Determinant ($\det(A)$): A scalar value giving scaled volume and singularity test. If $\det(A)=0$, $A$ is singular (noninvertible).
• Rank ($\operatorname{rank}(A)$): The dimension of the column (or row) space; number of linearly independent columns. Rank reveals degrees of freedom and whether linear systems have unique solutions.
• Trace ($\operatorname{trace}(A)$): Sum of diagonal elements, invariant under similarity transforms. For square $A$, $\operatorname{trace}(A)=\sum_i A_{ii}$.

Practical note: For numerical work, compute rank and inverse with tolerances; small determinants close to zero can indicate ill-conditioning.

🧩 Eigenvalues and Eigenvectors

For square matrix $A$, an eigenvalue $\lambda$ and corresponding eigenvector $v$ satisfy $Av=\lambda v$. The operation that computes them is often called eig. Eigenvalues provide insight into matrix behavior (stability, modes, diagonalization). For defective matrices, you may not get a full set of linearly independent eigenvectors; consider generalized eigenproblems or Schur decomposition for robustness.

When working numerically, eigenvalues can be complex even for real matrices; sorting and normalization of eigenvectors are common post-processing steps.

⚙️ Special Matrices

Common constructors useful for initialization and testing:

• zeros(n,m): $n\times m$ matrix of zeros.
• ones(n,m): $n\times m$ matrix of ones.
• eye(n): $n\times n$ identity matrix $I$ with ones on the diagonal.
• rand(n,m): random matrix with entries uniformly distributed in $(0,1)$.
• randn(n,m): random matrix with entries drawn from a standard normal distribution (mean 0, variance 1).

These are essential for building test cases, initializing algorithms, and setting up identity/scaling matrices.

🧾 Practical Tips & Numerical Considerations

• Prefer solving linear systems $Ax=b$ via decomposition (e.g., LU, QR) rather than computing $A^{-1}b$ explicitly. This improves accuracy and efficiency.
• Be aware of conditioning: a matrix with large condition number amplifies numerical errors. Use $\operatorname{cond}(A)$ to assess.
• Use appropriate tolerances when testing singularity or rank; floating-point arithmetic can make exact comparisons unreliable.
• For large or sparse matrices, use specialized sparse data structures and algorithms to reduce memory and computation costs.

✅ Summary

Understand how to construct vectors/matrices, index and slice, perform core operations, and compute key properties like inverse, determinant, rank, trace, and eigen-decomposition. Use special constructors like zeros, ones, eye, rand, and randn for practical tasks and testing.