Special Matrices: Identity, Diagonal, Symmetric

Linear Algebra for Machine Learning · Axiom Academy

Identity, Diagonal, Symmetric & More The identity matrix I is a square matrix with 1s on the diagonal and 0s elsewhere. It acts as the multiplicative identity: AI = IA = A. AI = A and IA = A for any matrix A All entries except those on the diagonal are zero. Diagonal matrices are easy to work with: multiplication, inversion, and determinants are simple. Multiplication is componentwise D⁻¹ exists if all diagonal entries are nonzero det(D) = product of diagonal entries A matrix A is symmetric if Aᵀ = A. This means aᵢⱼ = aⱼᵢ for all i, j. Can be diagonalized by orthogonal matrices Appear frequently in optimization and statistics A matrix A is skew-symmetric if Aᵀ = -A. This means aᵢⱼ = -aⱼᵢ for all i, j. The diagonal must be all zeros. Determinant is zero for odd-sized matrices Used in cross products and rotations All entries below the diagonal are zero: All entries above the diagonal are zero: Determinant = product of diagonal entries Easy to solve systems of equations A square matrix Q is orthogonal if QᵀQ = I. This means the columns are orthonormal (perpendicular unit vectors). Preserves lengths: ||Qx|| = ||x|| Preserves angles between vectors Rotation and reflection matrices are orthogonal

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