Final Study Flashcards
What is an Eigen Value?
Let matrix A be nn. Then the Eigen Value for A is a scalar /\ (Lambda) such that Avec = /*vec for some non-zero vector vec Exist in R^n.
If a matrix A, multiplied by a vector, is equal to a scalar multiplied by that same vector, then that scalar is called the Eigen value for A, and that vector is called an Eigen Vector for A.
What is an Eigen Vector?
Let matrix A be nn. Then the Eigen Vector for A is a vector vec, such that Avec = /*vec for some non-zero vector vec Exist in R^n.
If a matrix A, multiplied by a vector, is equal to a scalar multiplied by that same vector, then that scalar is called the Eigen value for A, and that vector is called an Eigen Vector for A.
How do you find Eigen Vectors?
Let A be nn. Let /\ be an Eigen Value for A. Let vec be any Eigen Vector for /. Then:
Avec=/*vec ==> Avec-/*vec=0vec
==>Avec-/*(invec)=0vec ==> Avec-(/*in)vec=0vec
Factor vec out==> (A-/*in)vec=0vec
Now we can see that vec is in the Nullspace(A-/*in), since it is a solution to the homogeneous equation. So this means that the set of non-zero vectors in the Nullspace, is also a set of Eigen Vectors FOR THAT SPECIFIC Eigen Value
So just find a basis for the Nullspace of (A-/*in), and you will have a set of Linearly Independent Eigen Vectors, FOR THAT SPECIFIC Eigen Value.
CAREFUL: If an Eigen value is negative, then the form will be (A+/*in)=0vec PLUS!!!
What is an Eigen Space?
Let A be an nn matrix with Eigen Value /\1. Then the set of all Eigen Vectors, including the 0vec are a subspace of R^n called the Eigen Space of A corresponding to /\1. This subspace is denoted E/\1, and is equal to Null(A-/\1in).
How do you calculate Eigen Values?
Let matrix A be n*n. Then /\ Exist in R^n is an Eigen Value for A if and only if det(A-/*in)=0
What is a Triangular matrix?
An n*n matrix A is called upper or lower triangular if it only has zeros above or below the main diagonal. Echelon from is an example of Upper Triangular. (Because the non-zero values are in the upper half).
Given a triangular matrix, what is a fast way to find it’s determinant?
Given a triangular matrix, you can simply multiply all the diagonal entries together. Their product is the determinant.
This can be verified by using co-factor expansion.
What is the Characteristic Polynomial?
Let A be n*n, the det(A-/*in) is a polynomial of degree n. This is called the Characteristic Polynomial and it’s roots are exactly Eigen Values of A.
What is a Diagonal Matrix?
It is an n*n Matrix that has ALL ZEROS above and below the main diagonal.
What are similar Matrices?
Let A and B be nn matrices. Then A is similar to B if there exists an nn invertible matrix P, such that
A=PB(P^-1).
Similar Matrices have the same Eigen Values, Determinant and Ranks.
What does it mean for a Matrix to be Diagonalizable?
An nn matrix A is diagnonalizable if it is similar to a diagonal matrix D. Meaning A=PD(P^-1), where P is an nn invertible matrix.
What is the Multiplicity of an Eigen Value?
Given Eigen value /\, it’s multiplicity, denoted m(/), is it’s multiplicity as a root of the characteristic polynomial.
i.e. if det(A-/*in)=(/-2)^2, then A has one eval, /=2, and m(2)=2. (because the polynomial is raised to the power of 2)
If an n*n matrix has an Eigen value of 0, what do we know?
Any matrix that has an Eigen Value of 0 IS NOT invertible.
For an n*n Matrix A, what can the sum of the Eigenspace dimensions tell us?
Given an n*n matrix A, if the sum of it’s Eigenspace dimensions do not equal n, then A is NOT diagonalizable.
