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MAT223 Lecture 10
MAT223 Lecture 10 Raw
MAT223 Lecture 10 Flashcards
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Completed Notes Status
- Completed insertions: 9
- Ambiguities left unresolved: 1
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Lecture Summary
- Central objective: Understand the process of finding eigenvalues and eigenvectors, and analyze the relationship between matrix invertibility and diagonalizability.
- Key concepts:
- Eigenvalue and Eigenvector computation: Finding roots of the Characteristic Polynomial
and solving the Homogeneous System . - Matrix Diagonalization (
): The order of eigenvalues in the diagonal matrix must match the order of corresponding basic eigenvectors in matrix . - Matrix Invertibility: A matrix is invertible if and only if
, , and the homogeneous system has only the trivial solution.
- Eigenvalue and Eigenvector computation: Finding roots of the Characteristic Polynomial
- Connections:
- There is no direct connection between a matrix being a Diagonalizable Matrix and being invertible; a matrix can be one, both, or neither.
- A square matrix is non-invertible if and only if
is an Eigenvalue of the matrix.
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TK Resolutions
- #tk: idk the above, doesn't make sense. (Regarding why a matrix having 0 as an eigenvalue means it is non-invertible, and vice-versa).
- Answer: A matrix
is non-invertible if and only if its determinant . The characteristic equation is . If we plug in , we get . Therefore, if , then is a valid root (eigenvalue). Conversely, if is an eigenvalue, then , which proves is non-invertible.
- Answer: A matrix
- #tk: idk the above, doesn't make sense. (Regarding why a matrix having 0 as an eigenvalue means it is non-invertible, and vice-versa).
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Practice Questions
- Remember/Understand:
- What is the required relationship between the matrices
and when diagonalizing a matrix as ? - What must be true about the eigenvalues of a matrix if the matrix is known to be non-invertible?
- What is the required relationship between the matrices
- Apply/Analyze:
- Given an upper-triangular
matrix, how can you immediately determine its characteristic polynomial and eigenvalues? - If a
matrix has distinct eigenvalues , prove whether it is necessarily diagonalizable.
- Given an upper-triangular
- Evaluate/Create:
- Construct an example of a
matrix that is invertible but not diagonalizable, and prove why it fails to be diagonalizable.
- Construct an example of a
- Remember/Understand:
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Challenging Concepts
- Matrix Invertibility vs. Diagonalizability:
- Why it's challenging: It is easy to assume that a matrix failing to have full rank (non-invertible) also fails to be diagonalizable, but the two properties are entirely independent.
- Study strategy: Memorize counter-examples for each case (e.g., the identity matrix is both; a shear matrix
is invertible but not diagonalizable; a zero matrix is diagonalizable but not invertible).
- Characteristic Polynomial of singular matrices:
- Why it's challenging: Understanding why
forces the constant term of the characteristic polynomial to be zero, tying back to the determinant. - Study strategy: Write out the characteristic polynomial
and evaluate it explicitly at to see that it equals .
- Why it's challenging: Understanding why
- Matrix Invertibility vs. Diagonalizability:
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Action Plan
- Immediate review actions:
- Practice and application:
- Deep dive study:
- Verification and integration:
- Immediate review actions:
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Footnotes