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# Special Matrices | Engineering Mathematics | GATE Preparation | ME

**Application**

Special matrices based on symmetry find their application in mathematical analysis. Eigen Values and Eigen vectors of special matrices display some important properties.

**Explanation**

To understand this concept, knowledge of matrix representation and transpose is required.

Symmetric Matrix: Transpose of a square matrix is equal to matrix itself.

Skew Symmetric Matrix: Transpose of a square matrix is equal to matrix itself with negation. All principal diagonal elements are equal to zero.

Non Sy

**Orthogonal Matrix:** Transpose of a square matrix is equal to its inverse. Orthogonal matrix can have determinant as one or minus one. All rotational matrices are orthogonal matrices with determinant equal to one. All rows (or Columns) satisfy orthonormality of vectors.

Hermitian Matrix: In case of complex square matrices, transpose of a conjugate is equal to matrix itself. All principal diagonal elements are equal to real.

**Skew Hermitian Matrix:** In case of complex square matrices, transpose of a conjugate is equal to matrix itself with negation. All principal diagonal elements are purely imaginary.

**Unitary Matrix:** In case of complex square matrices, transpose of a conjugate of a matrix is equal to its inverse. All rows ( or Columns) satisfy orthonormality of complex vectors.

**Faculty Keywords:** Symmetric, Skew Symmetric, orthogonal, hermitian, Skew Hermitian, Unitary, Rotation Matrix.

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