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 $y$? \begin{equation*} 5x^2 + 8 y ^2 – 4 xy \geq 0 \end{equation*} Were…Continue readingRepresenting a Quadratic Form Using a Matrix
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
The eigenvalues of a square matrix with real entries can be complex. For example, a quick calculuation will verify that the eigenvalues of $$A = \begin{pmatrix} 1&-2\\2&1\end{pmatrix}$$ are $1\pm2i$. But when the matrix is symmetric, the eigenvalues must be real.…Continue readingEigenvalues of Real Symmetric Matrices
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
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
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
