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Questions and Answers
What is a defining characteristic of an orthogonal matrix?
What is a defining characteristic of an orthogonal matrix?
- It has zero elements only above the diagonal.
- It has all non-zero elements.
- Its transpose is equal to its inverse. (correct)
- It can only be a rectangular matrix.
Which statement about triangular matrices is true?
Which statement about triangular matrices is true?
- They can only have zeros above the diagonal.
- They must be symmetric matrices.
- They are always orthogonal.
- They are square matrices with zeros either below or above the diagonal. (correct)
What does diagonalization of a matrix involve?
What does diagonalization of a matrix involve?
- Representing the matrix as a sum of identity matrices.
- Transforming the matrix to a rectangular form.
- Finding all eigenvalues and eigenvectors only.
- Writing the matrix in the form $PDP^{-1}$ where $D$ is a diagonal matrix. (correct)
Which property is preserved under orthogonal transformations?
Which property is preserved under orthogonal transformations?
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How is matrix multiplication characterized in terms of commutativity?
How is matrix multiplication characterized in terms of commutativity?
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What is the relationship between an eigenvector and eigenvalue of a matrix?
What is the relationship between an eigenvector and eigenvalue of a matrix?
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Which of the following correctly describes an orthogonal transformation?
Which of the following correctly describes an orthogonal transformation?
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For a vector $x$ and orthogonal matrix $Q$, what can be inferred about $||Qx||$?
For a vector $x$ and orthogonal matrix $Q$, what can be inferred about $||Qx||$?
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Study Notes
Triangular Matrices and Transpose
- Triangular matrices are square matrices with zero elements either below (lower triangular) or above (upper triangular) the main diagonal.
- A matrix is orthogonal if its transpose equals its inverse, i.e., ( Q^T = Q^{-1} ).
Diagonalization of Matrices
- A matrix is diagonalizable if it can be expressed in the form ( PDP^{-1} ), where ( D ) is a diagonal matrix.
- Orthogonal matrices are always diagonalizable.
Orthogonal Transformation
- An orthogonal transformation is a linear transformation that maintains both the dot product and the lengths of vectors.
- This transformation is represented by an orthogonal matrix, satisfying the condition ( Q^TQ = I ).
Linear Systems and Eigenvalues
- Eigenvectors are nonzero vectors that transform under a matrix's action by only scaling, characterized by the eigenvalue.
- Orthogonal transformations preserve the eigenvalues of a matrix, meaning the eigenvalues remain unchanged.
Matrix Addition, Subtraction, and Multiplication Rules
- Matrix addition and subtraction are performed element-wise, meaning corresponding elements are added or subtracted.
- Matrix multiplication is achieved by taking the dot product of rows from the first matrix with columns from the second.
- While matrix multiplication is associative, it isn't commutative, i.e., ( AB \neq BA ) in general.
- Orthogonal matrices preserve the lengths of vectors during multiplication, indicating ( ||Qx|| = ||x|| ) for any vector ( x ).
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