Eigenvalues and Eigenvectors in Linear Algebra
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ORTHONORMAL BASES; GRAM–SCHMIDT PROCESS
DEFINITION
A set of vectors in an inner product space is called an
orthogonal set
if all pairs of distinct vectors
in the set are orthogonal. An orthogonal set in which each vector has norm 1 is called
orthonormal
.
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Eigenvalues, Eigenvectors
Chapter 7
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7.1
EIGENVALUES AND EIGENVECTORS
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A
The determinant is always a polynomial p in
λ
p(
λ
)
, called the characteristic
polynomial of A.
EXAMPLE 3 Eigenvalues of an Upper Triangular Matrix
Find the eigenvalues of the upper triangular matrix
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If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of A are the
entries on the main diagonal of A.
EXAMPLE 4 Eigenvalues of a Lower Triangular Matrix
By inspection, the eigenvalues of the lower triangular matrix
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EXAMPLE 5 Eigenvectors and Bases for Eigenspaces
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are linearly independent, these vectors form a basis for the eigenspace corresponding
to
λ
=2
.
If
λ
=1
, then
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A square matrix A is invertible if and only if
λ
=0
is not an eigenvalue of A.
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The matrix A in Example 5 is invertible since it has eigenvalues
THEOREM 7.1.5
Equivalent Statements
If A is an nxn matrix, and if T
A
: R
n
R
n
is multiplication by A, then the following are equivalent.
(a) A is invertible.
(b) AX=0 has only the trivial solution.
(c) The reduced row-echelon form of A is I
n
.
(d) A is expressible as a product of elementary matrices.
(e) AX=b is consistent for every nx1 matrix b .
(f) AX=b has exactly one solution for every nx1 matrix b.
7.2
DIAGONALIZATION
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DEFINITION
A square matrix
A
is called
diagonalizable
if there is an invertible matrix
P
such that P
-1
AP is a
diagonal matrix; the matrix
P
is said to
diagonalize
A
.
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If A is an nxn matrix, then the following are equivalent.
(a)
A is diagonalizable.
(b)
A has n linearly independent eigenvectors.
EXAMPLE 1 Finding a Matrix
P
That Diagonalizes a Matrix
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EXAMPLE 2 A Matrix That Is Not Diagonalizable
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THEOREM 7.2.3
If an nxn matrix A has n distinct eigenvalues, then A is diagonalizable.
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THEOREM 7.2.4
Geometric and Algebraic Multiplicity If A is a square matrix, then
(a) For every eigenvalue of A, the geometric multiplicity is less than or equal to the algebraic
multiplicity.
(b) A is diagonalizable if and only if, for every eigenvalue, the geometric multiplicity is equal to the
algebraic multiplicity.
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Explore the concepts of eigenvectors and eigenvalues in linear algebra, from defining orthonormal bases and the Gram-Schmidt process to finding eigenvalues of upper triangular matrices. Learn the theorems and examples that showcase the importance of these concepts in matrix operations and transformations.
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Presentation Transcript
ORTHONORMAL BASES; GRAMSCHMIDT PROCESS DEFINITION A set of vectors in an inner product space is called an orthogonal set if all pairs of distinct vectors in the set are orthogonal. An orthogonal set in which each vector has norm 1 is called orthonormal. EXAMPLE Using the Gram Schmidt Process
Chapter 7 Eigenvalues, Eigenvectors
7.1EIGENVALUES AND EIGENVECTORS DEFINITION
This is called the characteristic equation of A The determinant is always a polynomial p in p( ), called the characteristic polynomial of A.
EXAMPLE 3 Eigenvalues of an Upper Triangular Matrix Find the eigenvalues of the upper triangular matrix
THEOREM 7.1.1 If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of A are the entries on the main diagonal of A. EXAMPLE 4 Eigenvalues of a Lower Triangular Matrix By inspection, the eigenvalues of the lower triangular matrix
EXAMPLE 5 Eigenvectors and Bases for Eigenspaces Find bases for the eigenspaces of are linearly independent, these vectors form a basis for the eigenspace corresponding to =2. If =1, then
Powers of a Matrix THEOREM 7.1.3 EXAMPLE 6 : Find eigenvalues of A7
Eigenvalues and Invertibility THEOREM 7.1.4 A square matrix A is invertible if and only if =0 is not an eigenvalue of A. EXAMPLE 7 Using Theorem 7.1.4 The matrix A in Example 5 is invertible since it has eigenvalues
THEOREM 7.1.5 Equivalent Statements If A is an nxn matrix, and if TA: Rn (a) A is invertible. (b) AX=0 has only the trivial solution. (c) The reduced row-echelon form of A is In. (d) A is expressible as a product of elementary matrices. (e) AX=b is consistent for every nx1 matrix b . (f) AX=b has exactly one solution for every nx1 matrix b. Rnis multiplication by A, then the following are equivalent.
(g) det(A) 0. (j) The column vectors of A are linearly independent. (k) The row vectors of A are linearly independent. (l) The column vectors of A span Rn. (m) The row vectors of A span Rn. (n) The column vectors of A form a basis for Rn. (o) The row vectors of A form a basis for Rn. (p) A has rank n. (q) A has nullity 0. (r) The orthogonal complement of the nullspace of A is Rn. (s) The orthogonal complement of the row space of A is {0}. (t) ATA is invertible. (u) =0 is not an eigenvalue of A.
7.2DIAGONALIZATION The Matrix Diagonalization Problem DEFINITION A square matrix A is called diagonalizable if there is an invertible matrix P such that P-1AP is a diagonal matrix; the matrix P is said to diagonalize A. THEOREM 7.2.1 If A is an nxn matrix, then the following are equivalent. (a) A is diagonalizable. (b) A has n linearly independent eigenvectors.
EXAMPLE 1 Finding a Matrix P That Diagonalizes a Matrix A Find a matrix P that diagonalizes
EXAMPLE 2 A Matrix That Is Not Diagonalizable Find a matrix P that diagonalizes
THEOREM 7.2.2 THEOREM 7.2.3 If an nxn matrix A has n distinct eigenvalues, then A is diagonalizable. EXAMPLE 3 Using Theorem 7.2.3
THEOREM 7.2.4 Geometric and Algebraic Multiplicity If A is a square matrix, then (a) For every eigenvalue of A, the geometric multiplicity is less than or equal to the algebraic multiplicity. (b) A is diagonalizable if and only if, for every eigenvalue, the geometric multiplicity is equal to the algebraic multiplicity. Computing Powers of a Matrix
