Final Study

Question 1

What is an Eigen Value?

Answer 1

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.

Question 2

What is an Eigen Vector?

Answer 2

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.

Question 3

How do you find Eigen Vectors?

Answer 3

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!!!

Question 4

What is an Eigen Space?

Answer 4

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).

Question 5

How do you calculate Eigen Values?

Answer 5

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

Question 6

What is a Triangular matrix?

Answer 6

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).

Question 7

Given a triangular matrix, what is a fast way to find it’s determinant?

Answer 7

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.

Question 8

What is the Characteristic Polynomial?

Answer 8

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.

Question 9

What is a Diagonal Matrix?

Answer 9

It is an n*n Matrix that has ALL ZEROS above and below the main diagonal.

Question 10

What are similar Matrices?

Answer 10

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.

Question 11

What does it mean for a Matrix to be Diagonalizable?

Answer 11

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.

Question 12

What is the Multiplicity of an Eigen Value?

Answer 12

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)

Question 13

If an n*n matrix has an Eigen value of 0, what do we know?

Answer 13

Any matrix that has an Eigen Value of 0 IS NOT invertible.

Question 14

For an n*n Matrix A, what can the sum of the Eigenspace dimensions tell us?

Answer 14

Given an n*n matrix A, if the sum of it’s Eigenspace dimensions do not equal n, then A is NOT diagonalizable.