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Linear Algebra Formulas
Browse 3 reviewed linear algebra formulas with concrete inputs, outputs, calculations, applicability checks, and LaTeX.
Linear Algebra
2×2 Matrix Determinant
\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bcCalculates the determinant of a two-by-two matrix.
Open formulaLinear Algebra
Cross Product Components
\mathbf a\times\mathbf b=\begin{bmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{bmatrix}Calculates the three-dimensional vector cross product.
Open formulaLinear Algebra
Vector Dot Product
\mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^{n}a_ib_i=\|\mathbf{a}\|\|\mathbf{b}\|\cos\thetaCombines two vectors into a scalar and relates their components to the angle between them.
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Linear Algebra
Cauchy–Schwarz Inequality
|\langle x,y\rangle|^2\le \langle x,x\rangle\langle y,y\rangleBounds the absolute inner product by the product of vector norms.
Open formulaLinear Algebra
Cramer's Rule for a 2×2 System
x=\frac{d_1b_2-b_1d_2}{a_1b_2-b_1a_2},\quad y=\frac{a_1d_2-d_1a_2}{a_1b_2-b_1a_2}Solves a nonsingular two-equation linear system using determinant ratios.
Open formulaLinear Algebra
Determinant as Eigenvalue Product
\det(A)=\prod_{i=1}^{n}\lambda_iRelates a square matrix determinant to its eigenvalues with multiplicity.
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Eigenvalue Equation
A\mathbf v=\lambda\mathbf vDefines an eigenvector and its eigenvalue.
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Euclidean Vector Norm
\|\mathbf v\|_2=\sqrt{\sum_i v_i^2}Calculates the Euclidean length of a vector.
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Inverse of a 2×2 Matrix
A^{-1}=\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}Computes the inverse of a nonsingular two-by-two matrix.
Open formulaLinear Algebra
Least-Squares Normal Equations
X^{\mathsf T}X\hat\beta=X^{\mathsf T}yCharacterizes ordinary least-squares coefficients through normal equations.
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Matrix Multiplication
(AB)_{ij}=\sum_k A_{ik}B_{kj}Defines each entry of a matrix product.
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Matrix Trace
\operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii}Adds the entries on the main diagonal of a square matrix.
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Quadratic Form
q(x)=x^TAxDefines a scalar quadratic expression associated with a square matrix.
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Three by Three Determinant
\det A=a(ei-fh)-b(di-fg)+c(dh-eg)Expands the determinant of a 3×3 matrix along the first row.
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Triangle Inequality for Norms
\|x+y\|\le \|x\|+\|y\|States that the direct displacement is no longer than a path through an intermediate point.
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Two by Two Matrix Inverse
A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}Finds the inverse of a nonsingular 2×2 matrix.
Open formulaLinear Algebra
Vector Projection
\operatorname{proj}_{\mathbf b}\mathbf a=\frac{\mathbf a\cdot\mathbf b}{\mathbf b\cdot\mathbf b}\mathbf bProjects one vector onto the direction of another.
Open formulaLinear Algebra
Vector Projection
\operatorname{proj}_{u}v=\frac{v\cdot u}{u\cdot u}uProjects one vector onto the direction of another.
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