Linear Algebra

Sheet 1

Vector Spaces, Eigenvalues & Decompositions

8 formulas

Key formulas

Rank of a Matrix ☆

\[ \text{Rank}(A) = \text{Rank}(A^T) \]

= largest order of a non-zero minor

Rank-Nullity Theorem ☆

\[ \text{rank}(A) + \text{nullity}(A) = n \]

n = number of columns of A

Characteristic Equation ☆

\[ \det(A - \lambda I) = 0 \]

roots are the eigenvalues

Eigenvalue Sum & Product ☆

\[ \sum \lambda_i = \text{trace}(A) \qquad \prod \lambda_i = \det(A) \]

Orthogonal Matrix ☆

\[ A^{-1} = A^T \]

Row-Echelon Solvability ☆

\[ \text{rank}([A]) = \text{rank}([A \mid b]) \]

consistent system of Ax = b

LU Decomposition ☆

\[ A = LU \]

L lower-triangular, U upper-triangular

Singular Value Decomposition ☆

\[ A = U \Sigma V^T \]

works for any m×n matrix, not just square ones

Test your recall

Can you recall the rank of a matrix formula?

Reveal formula

\[ \text{Rank}(A) = \text{Rank}(A^T) \]

GATE traps 4 traps

Singular means a zero eigenvalue

A singular matrix (det = 0) has at least one zero eigenvalue -- a matrix with all non-zero eigenvalues is always invertible.

Triangular/diagonal eigenvalues sit on the diagonal

For a triangular or diagonal matrix, the eigenvalues are exactly the diagonal entries -- no characteristic-equation work needed.

SVD isn't limited to square matrices

Eigendecomposition (A = PDP⁻¹) requires a square, diagonalizable matrix. SVD exists for every matrix, square or not -- that generality is exactly why it's used in dimensionality reduction on rectangular data.

Repeated eigenvalues aren't automatically bad

A repeated eigenvalue doesn't automatically mean a matrix is non-diagonalizable -- check whether geometric multiplicity equals algebraic multiplicity before concluding either way.