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2 cards from this section. Quick, good the night before a test.
Define algebraic multiplicity in your own words.
Think about the roots of a polynomial.
What does geometric multiplicity represent?
Consider how many independent vectors relate to an eigenvalue.
What does algebraic multiplicity refer to?
Think about how the characteristic polynomial is defined.
Is the geometric multiplicity ever greater than the algebraic multiplicity?
Recall the inequality established in this section.
Consider the 2x2 matrix A = | 1 2 | | 0 1 |. Determine the eigenvalues, their algebraic and geometric multiplicities, and discuss their implications for the matrix.
Examine the matrix closely using the determinant for λ.
Create a 3x3 matrix with an eigenvalue having an algebraic multiplicity of 3 but a geometric multiplicity of 1. Analyze the eigenvalue's implications on linear transformations.
Focus on the structure of rows and columns leading to repeated eigenvalues.
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