VM270 6350
Chapter 5: Eigenvalues and Eigenvectors
Matrices
Diagonal
Matrix with non-zero elements only along the diagonal of the matrix and zeroes everywhere else
Upper Triangular
2nd Corollary from Ch. 6: Supose V is a complex vector space and T c L(V). Then T has an upper-triangular matrix with respect to some orthonormal basis of V.
This corollary is essentially "cutting out the fat" of the previous two.
Corollary from CH. 6: If the above theorem is true, then T has an upper-triangular matrix with respect to some orthonormal basis of V.
Theorem: Suppose V is a complex vector space and T c L(V). Then T has an upper-triangular matrix with respect ot some basis of V.
Theorem: Every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue.
Tu = λu
The non-zero vector u c V is called the eigenvector of T (corresponding to λ).
null (T -λI) = The set of eigenvectors of T corresponding to λ.
The scalar λ c F is an eigenvalue of T c L(V) when u is a non-zero vector.
Recall: An Operator is a linear map function from a vector space to itself.
NOTE FOR PROF BARAKAT
I'm sorry if this mindmap is messy. After a while I grew tired of constantly switching between underscores and italics and all this would quickly aggravate my Carpal Tunnel, so at some point some of this typing is mostly in a way that is not necessarily the cleanest way of typing things, but the fastest and easy enough for me to understand. After all, these maps were mainly for our own benefit in grasping the subject, weren't they?
Also, the boundaries were included to highlight what I believed to be the most fundamental or difficult topics we've gone over so I could quickly find them.
Sorry for any inconvenience
Chapter 6: Inner-Product Spaces
Orthogonal Decomposition
u = av + w.
Set w = u - av
Take into account that = 0 because w is supposed to be orthogonal to av.
Find a value for a in terms of u and v.
Substitute the value into u = av + (u - av)
This gives you an Orthogonal Decomposition.
Note: Originally wrote much more than this, but Mindomo deleted the branch and couldn't redo it so this is sufficient for now.
Theorems
If U is a subspace of V, then
V equals the direct sum of U and the orthogonal complement of U.
The Gram-Schmidt procedure
This procedure essentialy allows you to transform a linearly independent set of vectors (v1,...,vm) into an orthonormal set (e1,...,em) of vectors in V, such that span (v1,...,vj) = span (e1, ..., ej) for j= 1, ..., m.
Parallelogram Equality
This is one of the easiest theorems that you realize you need to use during a proof/problem. If it uses u+v or u-v, there's a good chance you need to use this equality.
||u+v||^2 + ||u-v||^2 = 2 * (||u||^2 + ||v||^2)
Triangle Inequality
||u + v|| <= ||u|| + ||v||
Cauchy-Schwarz Inequality
If u,v c V, then || =< ||u|| ||v||
Pythagorean Theorem
||u + v||^2 = ||u||^2 + ||v||^2
Definitions
Conjugate transpose
Linear Functional
Orthogonal Projection
Norm
The length of a vector
Adjoint
For T c L(V,W), =
A function/linear map
Orthogonal complement of U
The set of all vectors in V that are orthogonal to every vector in U. Denoted as U^(Perpendicular symbol)
Orthogonal
= 0.
Basically, this is when u and v are perpendicular to each other.
Inner-Product Space
Euclidean Inner Product
Def: A vector space V along with an inner product on V
What is an Inner-Product?
Conjugate Symmetry
Homogeneity in first slot
Additivity in first slot
Definiteness
= 0 if and only if v=0
Positivity
>= 0 for all v c V
Formal Definition
A function that takes every order pair (u, v) c V to the number c F.
An Inner Product basically measures how much u points in the direction of v when u,v C V. It takes two vectors and produces a number.
Chapter 7: Operators on Inner-Product Spaces
Isometry
For S c L(V), ||Sv|| = ||v||
Def: An operator is an isometry if it preserves norms.
The Spectral Theorem
Overall Result
Whether F=R or F=C, every self-adjoint operator on V has a diagonal matrix with respect to some orthonormal basis.
Components
Real Spectral Theorem
Supposing V is a real inner-product space and T c L(V), V has an orthonormal basis consisting of eigenvectors of T if and only if T is self-adjoint.
Lemmas
2
Suppose T c L(V) is self-adjoint. Then T has an eigenvalue.
Important mostly for Real vector spaces
1
Suppose T c L(V) is self-adjoint. If a¸b cR are such that
(a)^2 < 4b, then
T^2 + aT + bI
is invertible.
Complex Spectral Theorem
Supposing V is a complex inner-product space and T c L(V), V has an orthonormal basis consisting of eigenvectors of T if and only if T is normal.
What is it?
This theorem characterizes operators on V, for which there is an orthonormal basis of V with respect to the operator's diagonal matrix, as either a normal or self-adjoint operator, depending on whether F=C or F=R.
Self-Adjoint and Normal Operators
Normal
Prop: T is normal if and only if
||Tv|| = ||T*v||
Definition
TT* = T*T
In other words, the operator commutes with its adjoint.
All self-adjoint operators are also normal.
Self-Adjoint
For T c L(V), T = T*
Chapter 3: Linear Maps
Matrix
Range
A.k.a. Image
Null Space
Proposition: If T c L(V,W), then null T is a subspace of V
The subset of V consisting of the vectors that T maps to 0.
Types of Linear Maps
From F^n to F^m
Backward shift
Multiplication by x^2
Integration
Differentiation
Identity Map
Iv = v
Zero Function
0v = 0
Linear Map
Ex: T: V -> W maps a vector in V to W.
Def: A function that "maps" a vector from one vector space onto another vector space
Properties of a Linear Map
Homogeneity
T(av) = a(Tv)
Additivity
T(u + v) = Tu + Tv
Vocabulary
Intersection (Ո)
Remember, it's the KFC.
The set of elements in set A and set B.
Union (U)
Remember, it's the stores on street A and B, including the KFC on the corner.
The set of elements in set A or set B.
Operator
Def: A linear map function from a vector space to itself.
Invariant
A subspace that gets mapped onto itself by an operator is said to be invariant.
Subjective
For T: V -> W, T is surjective if its range equals W.
Def: Onto
Injective
For T: V -> W, T is injective whenever u, v c V and Tu = Tv, we get u = v.
Def: One to one
Chapter 2:
Finite Dimensional
Vector Spaces
Isomorphic
Two vector spaces are isomorphic if there is an invertible linear map from one vector space onto the other one.
Invertible
A linear map T c L(V,W) is invertible if there is a linear map S c L(W,V) such that ST = I c V and TS = I c W.
Dimension
Bases
Linear Dependence and Independence
Linear Dependence
Linear Independence
Span
Linear Combination
Def: Composed of a list (v1, ..., v
Chapter 1: Vector Spaces
Direct Sum
Theorem
If U and W c V, then V is a direct sum of U and W if and only if V = U + W and U Ո W = {0}
When every vector in V can be represented as a unique linear combination of subspaces U and W, it is said that V is a Direct Sum of U and W.
Sum
A Sum is a linear combination of a vector in V using vectors in subspaces of V.
Notation/Example:
V = U + W when U, W c V.
Vector Spaces
Must have a addition function and a scalar multiplication function so the other properties may follow:
Distributative Properties
(a + b)u = au + bu for all a,b c F and all u c V
a (u + v) = au + av for all a c F and all u,v c V
1v = v for all v c V
For every v c V, there exists w c V such that v + w = 0.
Additive Identity
There exists an element 0 c V such that v + 0 = v for all v c V
(ab)v = a(bv) for all a,b c F and all v c V
(u + v) + w = u + (v + w) for all u,v,w c V
u + v = v+ u for all u,v c V
Definition
A set V with addition and scalar multiplication functions in V that allow the listed properties to hold.
Complex Numbers
Created this branch in case i needed to reference something.
Properties
Division
w/z = w * (1/z) for w,z c C with z not equal to 0.
z * (1/z) = 1 for z c C and z not equal to 0.
Subtraction
w - z = w + (-z) for w,z c C
z + (-z) = 0 for z c C
Distributative
Multiplicative Inverse
Additive Inverse
Identities
Associativity
Commutativity
Definition: C ={a+bi: a,b c R}
