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Graphs and Adjacency Matrices:(see MATLAB Project # 2)A graph is a set of points (called vertices or nodes)

and a set of lines (called edges or paths of length one) connecting some pairs of nodes.

Two nodes connected by an edge are said to be adjacent.

The Adjacency Matrix for a graph with nodes is an n x n matrix where

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Example. Graph of n=5 nodes with adjacency matrix

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4.7 Change of BasisQuestion. Suppose that and are two bases for

a vector space Let be a vector in How are and related?

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4.7 Change of BasisQuestion. Suppose that and are two bases for

a vector space Let be a vector in How are and related?

Answer:

where is the change-of-coordinates matrix from to

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Example. Let

If find

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Theorem. Let be two bases for a vector space Then there is a unique

matrix such that

is called the change-of-coordinates matrix from to

Furthermore,

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Question. Suppose that is known. How does one find

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If what is

1. (a)2. (b)3. (c)

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Example. Letbe two bases for and suppose that

1. Find 2. Find if

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5.1 Eigenvectors and Eigenvalues

Example. Let

Examine the images of and under multiplication by

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Definition. An eigenvector of a square matrix is a nonzero vector such that for some scalarA scalar is called an eigenvalue of if there is a nontrivial solution of such an is called an eigenvector of corresponding to

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Example. Show that 4 is an eigenvalue ofand find the corresponding eigenvectors.

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Let Is an

eigenvector of

1. Yes2. No

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The set of all solutions to is called the eigenspace of corresponding to the eigenvalue

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Example. Let An eigenvalue of is

Find a basis for the corresponding eigenspace.

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How are the eigenvalues of related to the eigenvalues of

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What are the eigenvalues of a triangular matrix?

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5.2 The Characteristic EquationRecall: eigenvector of corresponding to

nonzero.

How do we find the eigenvalues of

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To find the eigenvalues of

Characteristic polynomial:

Characteristic equation:

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Example. Find the eigenvalues of

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Example. Find the eigenvalues of

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Theorem. A is invertible if and only if _______ is not an eigenvalue of A.

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SimilarityFor square matrices and we say that is

similar to if there is an invertible matrix such that

Theorem. Similar matrices have the same determinant.

Proof.