Linear Algebra: Core Equations and Relations

Abstract

This article surveys foundational equations and structural relationships in linear algebra, focusing on matrix operations, determinants, eigenvalue problems, and diagonalization. These concepts form the backbone of computational linear algebra and its applications across engineering, data science, and physics. We present formal statements alongside intuitive interpretations and worked examples.

Background

Linear algebra provides a language for describing linear transformations and solving systems of equations. At its core are matrices—rectangular arrays of scalars—and the operations defined on them. Understanding how matrices interact, what properties they possess, and how to decompose them into simpler forms is essential for both theoretical understanding and practical computation.

The central question in much of linear algebra is: given a matrix $A$, what can we learn about its structure, invertibility, and behavior under repeated application? The answers involve determinants, eigenvalues, and eigenvectors.

Key Results

Matrix Multiplication

^{[matrix-multiplication]} defines the fundamental operation of combining two matrices. For matrices $A$ (size $m \times n$) and $B$ (size $n \times p$), the product $AB$ is computed entry-wise as:

$$(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$$

This operation is not commutative: in general, $AB \neq BA$. Matrix multiplication is essential for representing linear transformations and solving systems of equations. The non-commutativity reflects the fact that the order in which transformations are applied matters.

Determinants and Invertibility

The determinant is a scalar-valued function on square matrices that encodes critical information about invertibility and volume scaling. ^{[determinant-of-a-matrix]} establishes that for a $2 \times 2$ matrix,

$$\det(A) = ad - bc \quad \text{for} \quad A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

More generally, determinants can be computed via row reduction or cofactor expansion for larger matrices.

^{[determinant-properties]} and ^{[determinant-properties]} enumerate key properties:

1. Row swaps change the sign: $\det(A') = -\det(A)$
2. Scaling a row by scalar $c$ scales the determinant: $\det(cR) = c \cdot \det(A)$
3. The determinant of a triangular matrix equals the product of diagonal entries
4. Transposition preserves determinant: $\det(A^T) = \det(A)$
5. Adding a multiple of one row to another preserves the determinant

A matrix is invertible if and only if $\det(A) \neq 0$. A zero determinant indicates the transformation collapses space into a lower dimension, implying linear dependence among rows or columns.

Eigenvalues and Eigenvectors

^{[eigenvalues-of-a-matrix]} defines eigenvalues as scalars $\lambda$ satisfying the characteristic equation:

$$\det(A - \lambda I) = 0$$

Eigenvalues reveal how much a corresponding eigenvector is stretched or compressed under the transformation represented by $A$. They are crucial in stability analysis, principal component analysis, and understanding oscillatory behavior in dynamical systems.

Diagonalization

^{[diagonalizable-matrix]} and ^{[diagonalizable-matrix]} establish that a matrix $A$ is diagonalizable if there exist an invertible matrix $P$ and a diagonal matrix $D$ such that:

$$A = PDP^{-1}$$

Here, the columns of $P$ are linearly independent eigenvectors of $A$, and $D$ contains the corresponding eigenvalues on its diagonal. Diagonalization simplifies matrix powers and solving differential equations, since $A^n = PD^nP^{-1}$ and $D^n$ is trivial to compute.

Column Space and Null Space

^{[basis-of-column-space]} describes the column space $\text{Col}(A)$ as the subspace spanned by the columns of $A$. Its basis consists of the pivot columns in row echelon form, and its dimension equals the number of pivots (the rank of $A$).

^{[basis-of-null-space]} defines the null space $\text{Null}(A)$ as the solution set to $Ax = 0$. Its basis is found by solving the homogeneous system, and its dimension equals the number of free variables. Together, these spaces satisfy the rank-nullity relationship: $\text{rank}(A) + \text{nullity}(A) = n$ (the number of columns).

Matrix Equations

^{[matrix-equation-solution]} and ^{[matrix-inversion-formula]} demonstrate how to solve matrix equations by algebraic manipulation. For example, given invertible matrices $A$ and $B$, solving

$$B^{-1}(A - X) = AX$$

yields

$$X = (BA + I)^{-1}A$$

provided $BA + I$ is invertible. Such manipulations rely on properties of matrix multiplication and inversion, and the solution's existence depends on invertibility conditions.

Worked Examples

Example 1: Computing a 2×2 Determinant

Let $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$. By ^{[determinant-of-a-matrix]},

$$\det(A) = (2)(4) - (3)(1) = 8 - 3 = 5$$

Since $\det(A) \neq 0$, the matrix is invertible.

Example 2: Finding Eigenvalues

For the same matrix $A$, we solve $\det(A - \lambda I) = 0$ ^{[eigenvalues-of-a-matrix]}:

$$\det\begin{pmatrix} 2-\lambda & 3 \\ 1 & 4-\lambda \end{pmatrix} = (2-\lambda)(4-\lambda) - 3 = \lambda^2 - 6\lambda + 5 = 0$$

Factoring: $(\lambda - 1)(\lambda - 5) = 0$, so $\lambda_1 = 1$ and $\lambda_2 = 5$.

Example 3: Diagonalization

With eigenvalues $\lambda_1 = 1$ and $\lambda_2 = 5$, we find corresponding eigenvectors and form $P$ and $D$. If $P = \begin{pmatrix} 3 & 1 \\ 1 & 1 \end{pmatrix}$ and $D = \begin{pmatrix} 1 & 0 \\ 0 & 5 \end{pmatrix}$, then by ^{[diagonalizable-matrix]}, $A = PDP^{-1}$. Computing $A^{10}$ becomes $A^{10} = PD^{10}P^{-1}$, where $D^{10} = \begin{pmatrix} 1 & 0 \\ 0 & 5^{10} \end{pmatrix}$ is trivial.

References

• ^{[matrix-multiplication]}
• ^{[determinant-of-a-matrix]}
• ^{[determinant-properties]}
• ^{[determinant-properties]}
• ^{[determinant-of-a-matrix]}
• ^{[eigenvalues-of-a-matrix]}
• ^{[diagonalizable-matrix]}
• ^{[diagonalizable-matrix]}
• ^{[basis-of-column-space]}
• ^{[basis-of-null-space]}
• ^{[matrix-equation-solution]}
• ^{[matrix-inversion-formula]}

AI Disclosure

This article was drafted with the assistance of an AI language model based on personal class notes in Zettelkasten format. All mathematical statements and definitions are sourced from the cited notes and reflect standard linear algebra pedagogy. The article has been reviewed for technical accuracy and clarity.