Vector Algebra
- is a scalar. is an matrix.
Matrix can be
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Symmetric
- Positive-definite
- Positive-semi-definitie
- Negative-definite
- Negative-semi-definite
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Diagonal: entries outside the main diagonal are all zero
- Diagonalizable: is diagonalizable iff exists invertible matrix where is a diagonal matrix
- Inverse of a diagonalizable matrix is also diagonalizable
- Orthogonally diagonalizable: is orthogonally diagonizable iff exists orthogonal matrix where
- Diagonalizable: is diagonalizable iff exists invertible matrix where is a diagonal matrix
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Invertible vs Singular
- Invertible: There exists an inverse = determinant is non-zero
- The inverse of a symmetric matrix is also symmetric
- Singular: There does not exist an inverse
- Invertible: There exists an inverse = determinant is non-zero
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Orthogonal: is orthogonal iff
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Rank:= dimension of the vector space generated by its columns
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Identity Matrix
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Determinant: scalar value that determines if the matrix has a determinant
Matrix Algebra Identities
- Non-commutative
- Commutative only with scalars
- Distributive (w.r.t. matrix addition)
- Associative
- Computational complexity depends on which you multipliy first: Matrix Chain Multiplication
- Transpose-distribution
- Inverse-distribution
- Transpose-Inverse