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Symmetric matrices

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Definition of symmetric matrix
Symmetric matrices are square.
Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal.
Definition of orthogonally diagonalizable matrix
A matrix is orthogonally diagonalizable if and only if it is symmetric.
The spectral theorem for symmetric matrices
Formula for the spectral decomposition for a symmetric matrix
Definition of skew-symmetric matrix

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Symmetric matrices are square.

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Definition of symmetric matrixClose match
Symmetric matricesClose match
Definition of square matrixClose match
Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.Close match
Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.Close match
Definition of skew-symmetric matrixClose match
Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.Close match
Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).Close match
Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.Close match
Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.Close match
Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.Close match
Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.Close match
A matrix is orthogonally diagonalizable if and only if it is symmetric.Close match
Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.Close match
Equivalence theorem for nonsingular matrices: the matrix A has an inverse.Close match
Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.Close match
Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).Close match
Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).Close match
Equivalence theorem for nonsingular matrices: the matrix A has rank n.Close match
Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).Close match

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