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Chapter 7: Q40E (page 383)

TRUE OR FALSE

If a matrix is diagonalizable, then the algebraic multiplicity of each of its eigenvalues 位 must equal the geometric multiplicity of 位.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define Eigenvalue

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalues, often denoted by 位.

02

Explanation for the statement

According to Theorem 7.5.4., an n x n matrix has n eigenvalues, counted with their algebraic multiplicities. If there exists an eigenvalue 位 such gemu(位) < almu(位) , then the sum of geometric multiplicities of all eigenvalues is less than n, so there is no eigen basis, thus the matrix is not diagonalizable.

So, if a matrix is diagonalizable, then the algebraic and geometric multiplicity, of each eigenvalue 位, are the same.

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[010001100]

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