Linear Combinations – An introductory linear algebra blog

Representing a Quadratic Form Using a Matrix

March 18, 2021October 10, 2021greg.mayer

To motivate the use of linear algebra to study quadratic forms, consider the following problem. Does this inequality hold for all real values of $x$ and $y$ ? $5 x^{2} + 8 y^{2} - 4 x y \geq 0$ Were…Continue readingRepresenting a Quadratic Form Using a Matrix

Symmetric Matrices and Orthogonal Diagonalization

March 5, 2021September 18, 2021greg.mayer

Many algorithms rely on a type of matrix which are a type of matrix that is equal to its transpose. In other words, if matrix $A$ satisfies $A = A^{T}$ , then $A$ is symmetric. Note that if a matrix is symmetric, the…Continue readingSymmetric Matrices and Orthogonal Diagonalization

Eigenvalues of Real Symmetric Matrices

February 27, 2021October 10, 2021greg.mayer

The eigenvalues of a square matrix with real entries can be complex. For example, a quick calculuation will verify that the eigenvalues of $A = (\begin{matrix} 1 & - 2 \\ 2 & 1 \end{matrix})$ are $1 \pm 2 i$ . But when the matrix is symmetric, the eigenvalues must be real.…Continue readingEigenvalues of Real Symmetric Matrices

The LU Factorization

February 20, 2021greg.mayer

To solve a linear system of the form $A \vec{x} = \vec{b}$ we could use row reduction or, in theory, calculate $A^{- 1}$ and use it to determine $\vec{x}$ with the equation $\vec{x} = A^{- 1} \vec{b}$ But…Continue readingThe LU Factorization

Applying Matrix Multiplication to 2D Computer Graphics

February 7, 2021greg.mayer

Linear transformations are often used in computer graphics to simulate the motion of an object. They can be computed using a matrix-vector product of the form $T (\vec{x}) = A \vec{x}$ where $\vec{x}$ is a vector that represents a…Continue readingApplying Matrix Multiplication to 2D Computer Graphics

The Leontif Input-Output Model

January 29, 2021October 10, 2021greg.mayer

An input–output model are used in economics to model the inter-dependencies between different sectors of an economy. Wassily Leontief (1906–1999) is credited with developing the type of analysis that we explore in this chapter. His work on this model earned…Continue readingThe Leontif Input-Output Model

What Are Block Matrices? An Introduction to Block, or Partitioned, Matrices

January 8, 2021February 14, 2021greg.mayer

When dealing with large matrices that have a known structure, it can be more convenient to express the matrix in terms of what is known as a block, or partitioned, matrix. What is a Partitioned Matrix? This matrix: $$A =…Continue readingWhat Are Block Matrices? An Introduction to Block, or Partitioned, Matrices

Why are Eigenvectors from Distinct Eigenvalues Linearly Independent?

December 9, 2020January 8, 2021greg.mayer

In this post we will give a proof of the following theorem. Theorem If ${\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{r}$ are eigenvectors of matrix $A$ that correspond to distinct eigenvalues $λ_{1}, λ_{2}, \dots, λ_{r}$ , then the set…Continue readingWhy are Eigenvectors from Distinct Eigenvalues Linearly Independent?

Diagonalizability and Invertibility of a Matrix

July 16, 2020January 8, 2021greg.mayer

Is there a relationship between invertibility of a matrix and whether it can be diagonalized? For example, if a matrix is not invertible, can the matrix still be diagonalized? Before we explore these relationships, let’s give definitions for the invertibility…Continue readingDiagonalizability and Invertibility of a Matrix

Do Row Operations Change the Column Space of a Matrix?

July 2, 2020January 9, 2021greg.mayer

Row operations can change the column space of a matrix. But why might they do so, and do row operations always change the column space? Let’s recall a few definitions before we explore these questions. The Column Space of a…Continue readingDo Row Operations Change the Column Space of a Matrix?