Notes 1 and Notes 2(9-02-2026)

Covers the end of Notes 1 and the Beginning of Notes 2 highlighted (Refer to the lecture for proofs))

Equivalence

We say that matrices A and B in R^nxm are equivalent if there exists invertible matrices S₁ in R^nxn and S₂ in R^mxm such that we can form A as A = S₁BS₂

Similarity

We say that matrices A and B in R^nxn are similar if there exists an invertible matrix S in R^nxn such that A=SBS^-1

Congruence

We say that matrices A and B in R^nxn are congruent if there exists an invertible matrix S in R^nxn such that A=SBS^T

Orthogonal Similarity

A and B are orthogonally similar if they are both Similar and Congruent

The Characteristic Polynomial

Given a matrix A in R^nxn, The polynomial formed when you take the det(lambda*I-A)

Eigenvalues

The roots of the characteristic polynomial, are complex numbers

spec(A)

The set of all eigenvalues of matrix A

Algebraic multiplicity

Number of times a single eigenvalue is the root of the characteristic equation

Simple Matrix

The eigenvalues of the matrix all have a multiplicity of 1

Eigenvectors

x in Rⁿ is an eigenvector of A in R^nxn if (lambda*I-A)x = 0.

Geometric Multiplicity

How many linearly independent eigenvectors are associated with that particular eigenvalue lambda