Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q37E (page 338)

consider an eigenvalue $位_{0}$ of an $n 脳 n$ matrix A. we are told that the algebraic multiplicity of exceeds 1.Show that $f' (位_{0}) = 0$ (i.e.., the derivative of the characteristic polynomial of A vanishes are $位_{0}$ ).

Short Answer

Expert verified

Consider an eigenvalue.

If an eigenvalue $位_{0}$ of is of algebraic multiplicity greater than ,

it's at least 2.

$f' (位) = 2 f (位) (位 - 位_{0}) + f' (位) (位 - 位_{0}) 2 f' (位_{0}) = 0$

Step by step solution

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.

If an Eigenvalue $位_{0}$ of is of algebraic multiplicity greater than 1,

it's at least 2.

So, the characteristic polynomial is,

definition function

A function is a relationship between a set of inputs that each have one output.

A function is now, declare the values.

where is a function:

now,

. It is $f' (位) = 2 f (位) (位 - 位_{0}) + f' (位) (位 - 位_{0}) 2 f' (位_{0}) = 0$

Hence,

$f' (位) = 2 f (位) (位 - 位_{0}) + f' (位) (位 - 位_{0}) 2 f' (位_{0}) = 0$

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91影视

consider an eigenvalue $位_{0}$ of an $n 脳 n$ matrix A. we are told that the algebraic multiplicity of exceeds 1.Show that $f' (位_{0}) = 0$ (i.e.., the derivative of the characteristic polynomial of A vanishes are $位_{0}$ ).

Short Answer

Step by step solution

definition of eigenvalues

definition function

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91影视

consider an eigenvalue 位0of ann脳nmatrix A. we are told that the algebraic multiplicity of exceeds 1.Show thatf'(位0)=0(i.e.., the derivative of the characteristic polynomial of A vanishes are位0).

Short Answer

Step by step solution

definition of eigenvalues

definition function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Statistics

Geometry

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

consider an eigenvalue $位_{0}$ of an $n 脳 n$ matrix A. we are told that the algebraic multiplicity of exceeds 1.Show that $f' (位_{0}) = 0$ (i.e.., the derivative of the characteristic polynomial of A vanishes are $位_{0}$ ).