1 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
Transcript
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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.