Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

suppose a certain $4 \times 4$ matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

A is a $4 \times 4$ matrix its characteristic polynomial is of degree 4, which means it is .

$(λ - λ_{1}) (λ - λ_{2}) (λ - λ_{3}) (λ - λ_{4})$

role="math" localid="1659586683331" $A = \{\begin{matrix} 17 & 0 & 0 & 0 \\ 0 & 46 & 0 & 0 \\ 0 & 0 & 46 & 0 \\ 0 & 0 & 0 & 17 \end{matrix}\}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: definition of polynomial

A polynomial is a mathematical expression made up of indeterminates and coefficients that only involves the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

$4 \times 4$ Matrix A has two distinct real Eigenvalues:

If A is a $4 \times 4$ matrix, its characteristic polynomial is of degree , which means it is $(λ - λ_{1}) (λ - λ_{2}) (λ - λ_{3}) (λ - λ_{4})$ .

If we want two distinct eigenvalues, then we have two cases:

The two eigenvalues are of algebraic multiplicities 1 and 3 respectively.

Example is:

$A = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{matrix}]$

Step 2: definition of eigenvalues

The eigenvalue is a number that indicates how much variance exists in the data in that direction; in the example above, the eigenvalue is a number that indicates how spread out the data is on the line.

Both eigenvalues are of algebraic multiplicity 2

Example is:

$A = \{\begin{matrix} 17 & 0 & 0 & 0 \\ 0 & 46 & 0 & 0 \\ 0 & 0 & 46 & 0 \\ 0 & 0 & 0 & 17 \end{matrix}\}$

Hence,

$A = \{\begin{matrix} 17 & 0 & 0 & 0 \\ 0 & 46 & 0 & 0 \\ 0 & 0 & 46 & 0 \\ 0 & 0 & 0 & 17 \end{matrix}\}$

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Linear Algebra With Applications

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Short Answer

suppose a certain $4 \times 4$ matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

Step by Step Solution

Step 1: definition of polynomial

Step 2: definition of eigenvalues

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Linear Algebra With Applications

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Short Answer

suppose a certain 4×4 matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

Step by Step Solution

Step 1: definition of polynomial

Step 2: definition of eigenvalues

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suppose a certain $4 \times 4$ matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.