Linear Algebra Final Exam Review

Positive Definite

Positive Semidefinite

Eigen Value Test

Energy Test

Cholesky Factorization

Leading Determinant Test

Pivot Test

A^T Cholesky

A Cholesky

A^TA Cholesky

D

Rows of New U

*1’s in diagonal, lower triangular

L

*1’s in diagonal, upper triangular

Singular Value Decomposition (SVD)

Use to build U

Use to build V

Singular Values

Number of Singular Values

Use to build Σ

For Amxn, size of U is

For Amxn, size of V is

For Amxn, size of Σ is

To find Ui/Vi

Finding Ui via Vi

Find Vi via Ui

Find remaining Vi

Find remaining Ui

Complete the Square Formula

A is diagonalizable iff

Diagonalizable Form

What builds the matrix x as its columns?

Λ matrix

If A is diagonalizable, then det(A) =

If A is diagonalizable, then trace(A) =

trace(A+B)

trace(AB)

trace(kA)

trace(A^T)

trace(0matrix)

If A is diagonalizable, then A^ℓ

Any upper triangular matrix has

Similar matrices have the same

Spectral Theorem Formula for A Symmetric Matrix

Qi

*Where Ui = eigen vector

How to Build Q

det(I) where I is the identity matrix

Even Number of Row Exchanges

Odd Number of Row Exchanges

Constant Property

Split Property for a1+a2 b1+b2 in first row

det(A) if A has 2 same rows

det(A) if A has 0 row(s)

det(A) if A is triangular

*product of pivots/diagonal

A is invertible means

2×2 Determinant

3×3 Determinant

*cofactor expansion

Cofactor Matrix Entry Cij

Mij is Matrix A with

AC^T

By Cramer’s Rule, x^k =

A^k

*with b

det(AB)

det(A^P)

det(kA)

det(A^T)

Area of Triangle ABC

Area of Pyramid ABCD

AB, AC, AD Vectors

Eigen Value and Vector

Formula to find Eigen Value

Formula to find Eigen Vector AFTER finding Eigen Value

Trace(A)

Eigen Pair of A + kI

Eigen Pair of kA

Eigen Pair of A^-1

Eigen Pair of A^P

Eigen Pair of A^T

Characteristic Equation

Algebraic Multiplicity AM(λ)

Geometric Multiplicity GM(λ)

det(A) via Cofactor Expansion along Column j

det(A) via Cofactor Expansion along Row i

Hadamard’s Inequality

Characteristic Polynomial for a 2×2 Matrix

If A2×2, then trace(A) =

*Sum of eigen values for Apxp

If A2×2, then det(A) =

*Product of eigen values for Apxp

A^-1 from Cofactor Expansion