Transposed and symmetric matrices

Transposed and symmetric matrices

Transposed Matrices

• A transposed matrix is created by interchanging the rows and columns of a given matrix.
• If you have a matrix A, its transpose is denoted as Aᵀ.
• To calculate the transpose of a matrix, make the first row the first column, the second row the second column, and so on for all rows.

Properties of Transposed Matrices

• The transpose of a transpose is the original matrix. So, (Aᵀ)ᵀ = A.
• The transpose of a sum of matrices is equal to sum of their transposes. Meaning, (A + B)ᵀ = Aᵀ + Bᵀ.
• The transpose of a product of matrices equals the product of their transposes in the reverse order. So, (AB)ᵀ = BᵀAᵀ.

Symmetric Matrices

• A matrix is called symmetric if it is equal to its transpose. Formally, A is a symmetric matrix if A = Aᵀ.
• Symmetric matrices always have real eigenvalues, even if the matrix itself consists complex numbers.
• If a matrix is symmetric, the row space and column space of the matrix are the same.

Properties of Symmetric Matrices

• The sum of two symmetric matrices produces another symmetric matrix.
• If A is a symmetric matrix and K is any constant, then KA is symmetric.
• The product of two symmetric matrices is not necessarily symmetric.
• If A is a symmetric matrix, then for any real number r, rA is also symmetric.
• The inverse of a symmetric matrix is also symmetric (provided the inverse exists).