Module 1c: Linear Algebra Formula Sheet - ASTR 596: Modeling the Universe

Module 0a Quick Formula Sheet¶

Essential Vector Operations¶

Dot Product:

\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z = |\vec{a}||\vec{b}|\cos\theta

(1)

Physical Applications:

Work: $W = \vec{F} \cdot \vec{d}$
Power: $P = \vec{F} \cdot \vec{v}$
Projection of $\vec{a}$ onto $\vec{b}$ : $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$

Cross Product:

\vec{a} \times \vec{b} = \begin{pmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{pmatrix}

(2)

Magnitude: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

Physical Applications:

Angular momentum: $\vec{L} = \vec{r} \times \vec{p}$
Torque: $\vec{\tau} = \vec{r} \times \vec{F}$
Area of parallelogram: $A = |\vec{a} \times \vec{b}|$

Vector Norms and Properties:

Magnitude: $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$
Unit vector: $\hat{v} = \vec{v}/|\vec{v}|$
Orthogonal if: $\vec{a} \cdot \vec{b} = 0$

Essential Matrix Operations¶

Basic Operations:

Matrix multiplication: $(AB)_{ij} = \sum_k A_{ik}B_{kj}$
Transpose: $(A^T)_{ij} = A_{ji}$
Identity: $I\vec{x} = \vec{x}$ for all $\vec{x}$

Determinant:

2×2 Matrix:

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

(3)

Properties:

$\det(AB) = \det(A)\det(B)$
$\det(A^T) = \det(A)$
$\det(A^{-1}) = 1/\det(A)$
Singular if $\det(A) = 0$

Matrix Inverse:

2×2 Matrix:

\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

(4)

General Properties:

$(AB)^{-1} = B^{-1}A^{-1}$
$(A^T)^{-1} = (A^{-1})^T$
$(A^{-1})^{-1} = A$

Eigenvalues and Eigenvectors¶

Definition:

A\vec{v} = \lambda\vec{v}

(5)

Characteristic Equation:

\det(A - \lambda I) = 0

(6)

Properties:

Trace: $\text{tr}(A) = \sum_i A_{ii} = \sum_i \lambda_i$
Determinant: $\det(A) = \prod_i \lambda_i$
For symmetric matrices: all eigenvalues are real

Numerical Stability Quick Checks¶

Condition Number¶

\kappa(A) = ||A|| \cdot ||A^{-1}|| = \frac{\sigma_{\max}}{\sigma_{\min}}

(7)

Interpretation:

$\kappa < 10$ : Excellent
$10 < \kappa < 10^3$ : Good
$10^3 < \kappa < 10^6$ : Use caution
$\kappa > 10^6$ : Ill-conditioned

Machine Epsilon¶

Float64: $\epsilon \approx 2.2 \times 10^{-16}$
Float32: $\epsilon \approx 1.2 \times 10^{-7}$

Python/NumPy Quick Reference¶

# Vectors
np.dot(a, b)              # Dot product
np.cross(a, b)            # Cross product
np.linalg.norm(v)         # Vector magnitude

# Matrices
A @ B                     # Matrix multiplication
A.T                       # Transpose
np.linalg.inv(A)          # Inverse (avoid!)
np.linalg.solve(A, b)     # Solve Ax = b (preferred)
np.linalg.det(A)          # Determinant
np.linalg.eig(A)          # Eigenvalues & eigenvectors
np.linalg.cond(A)         # Condition number

# Numerical fixes
A_sym = (A + A.T) / 2     # Force symmetry
A_reg = A + eps * I       # Regularization

Module 0b Quick Formula Sheet¶

Positive Definite Matrices¶

Definition A symmetric matrix $A$ is positive definite if:

\vec{x}^T A \vec{x} > 0 \quad \forall \vec{x} \neq \vec{0}

(8)

Tests for Positive Definiteness:

All eigenvalues > 0
All leading principal minors > 0
Cholesky decomposition exists
Can write as $A = B^TB$ for invertible $B$

Quadratic Forms:

Q(\vec{x}) = \vec{x}^T A \vec{x} = \sum_{i,j} A_{ij} x_i x_j

(9)

Physical Examples:

Kinetic energy: $T = \frac{1}{2}\vec{v}^T M \vec{v}$
Potential energy: $V = \frac{1}{2}\vec{x}^T K \vec{x}$

Covariance and Statistical Matrices¶

Covariance:

\text{Cov}(X,Y) = E[(X-\mu_X)(Y-\mu_Y)] = E[XY] - E[X]E[Y]

(10)

Correlation:

\rho_{XY} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}

(11)

Covariance Matrix:

\Sigma_{ij} = \text{Cov}(X_i, X_j)

(12)

Properties:

Diagonal: $\Sigma_{ii} = \text{Var}(X_i)$
Symmetric: $\Sigma_{ij} = \Sigma_{ji}$
Positive semi-definite: all eigenvalues ≥ 0

Sample Covariance:

\Sigma = \frac{1}{n-1} \sum_{i=1}^n (\vec{x}_i - \bar{\vec{x}})(\vec{x}_i - \bar{\vec{x}})^T

(13)

Or in matrix form:

\Sigma = \frac{1}{n-1} X_c^T X_c

(14)

where $X_c$ is centered data matrix

Multivariate Gaussian Distribution¶

Probability Density:

p(\vec{x}) = \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}} \exp\left(-\frac{1}{2}(\vec{x}-\vec{\mu})^T\Sigma^{-1}(\vec{x}-\vec{\mu})\right)

(15)

Mahalanobis Distance:

d_M(\vec{x}) = \sqrt{(\vec{x}-\vec{\mu})^T\Sigma^{-1}(\vec{x}-\vec{\mu})}

(16)

Log-likelihood (numerical stability)

\log p(\vec{x}) = -\frac{n}{2}\log(2\pi) - \frac{1}{2}\log|\Sigma| - \frac{1}{2}(\vec{x}-\vec{\mu})^T\Sigma^{-1}(\vec{x}-\vec{\mu})

(17)

Matrix Decompositions¶

Cholesky Decomposition For positive definite $A$ :

A = LL^T

(18)

where $L$ is lower triangular with positive diagonal

Applications:

Solving: $A\vec{x} = \vec{b}$ via $L\vec{y} = \vec{b}$ , then $L^T\vec{x} = \vec{y}$
Sampling: If $\vec{z} \sim N(0,I)$ , then $\vec{\mu} + L\vec{z} \sim N(\vec{\mu}, \Sigma)$

Singular Value Decomposition (SVD)

A = U\Sigma V^T

(19)

Components:

$U$ : Left singular vectors (orthonormal)
$\Sigma$ : Singular values (diagonal, non-negative)
$V^T$ : Right singular vectors (orthonormal)

Pseudoinverse:

A^+ = V\Sigma^+ U^T

(20)

where $\Sigma^+$ inverts non-zero singular values

Advanced Topics¶

Matrix Exponential

e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots

(21)

Solution to linear ODE:

\frac{d\vec{x}}{dt} = A\vec{x} \implies \vec{x}(t) = e^{At}\vec{x}(0)

(22)

Jacobian Matrix

J_{ij} = \frac{\partial f_i}{\partial x_j}

(23)

Linear approximation:

\vec{f}(\vec{x} + \delta\vec{x}) \approx \vec{f}(\vec{x}) + J\delta\vec{x}

(24)

Schur Complement For block matrix $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ :

S = A - BD^{-1}C

(25)

Matrix Norms¶

Frobenius Norm

||A||_F = \sqrt{\sum_{i,j} |a_{ij}|^2} = \sqrt{\text{tr}(A^T A)}

(26)

Spectral Norm

||A||_2 = \max_{||\vec{x}||=1} ||A\vec{x}|| = \sigma_{\max}

(27)

Numerical Stability Patterns¶

Safe Numerical Practices:

# Cholesky with automatic regularization
def safe_cholesky(A, jitter=1e-6):
    try:
        return np.linalg.cholesky(A + jitter * np.eye(len(A)))
    except:
        eigvals, eigvecs = np.linalg.eigh(A)
        eigvals = np.maximum(eigvals, jitter)
        A_fixed = eigvecs @ np.diag(eigvals) @ eigvecs.T
        return np.linalg.cholesky(A_fixed)

# Work in log space
log_prob = multivariate_normal.logpdf(x, mu, Sigma)

# Stable parameterization
log_param = optimizer.param
param = np.exp(log_param)  # Always positive

# Standardize data
X_std = (X - X.mean(axis=0)) / X.std(axis=0)

Common Fixes:

Problem	Solution
Cholesky fails	Add jitter: `A + 1e-6 * I`
Not symmetric	Force: `(A + A.T) / 2`
Ill-conditioned	Regularize diagonal
Probability underflow	Use log space
Negative eigenvalues	Threshold: `np.maximum(eigvals, 0)`

Python/SciPy Quick Reference¶

# Statistical
np.cov(X.T)                          # Covariance matrix
scipy.stats.multivariate_normal      # Multivariate Gaussian
.logpdf(x, mean, cov)                # Log probability

# Decompositions
np.linalg.cholesky(A)                # Cholesky (A must be PD)
np.linalg.svd(A)                     # SVD
scipy.linalg.expm(A)                 # Matrix exponential

# Checks
np.linalg.eigvalsh(A).min() > 0      # Is positive definite?
np.allclose(A, A.T)                  # Is symmetric?
np.linalg.cond(A)                    # Condition number

# Solving systems
np.linalg.solve(A, b)                # When A is square
np.linalg.lstsq(A, b)                # Least squares
np.linalg.pinv(A)                    # Pseudoinverse