Symmetric and Skew-Symmetric Matrices

Symmetric Matrix

A square matrix \(A\) is called symmetric if \(A=A^{T}\), i.e., \(a_{ji}=a_{ij}, \forall \text{ }i, j\).

Skew-Symmetric Matrix

A square matrix \(A\) is called skew-symmetric if \(A=-A^{T}\), i.e., \(a_{ji}=-a_{ij}, \forall \text{ }i, j\).

Hermitian and Skew-Hermitian Matrices

Hermitian Matrix

A complex square matrix \(A\) is called Hermitian if \(A=\overline{A^{T}}=A^{H}\), i.e., \(a_{ij}=\overline{a_{ji}}, \forall \text{ }i, j\).

Skew-Hermitian Matrix

A complex square matrix \(A\) is called skew-Hermitian if \(A=- \overline{A^{T}} =-A^{H}\), i.e., \(a_{ij}=-\overline{a_{ji}}, \forall \text{ }i, j\)

Orthogonal and Unitary Matrices

Unitary Matrix

A matrix \(A\) is called unitary if \(AA^H=I\).

Orthogonal Matrix

A matrix \(A\) is called orthogonal if \(AA^T=I\).

PYQs

Symmetric and Skew-Symmetric Matrices

1) When is a square matrix \(A\) said to be congruent to a square matrix \(B\)? Prove that every matrix congruent to skew-symmetric matrix is skew symmetric.

[2001, 15M]

Hermitian and Skew-Hermitian Matrices

1) Prove that Eigen values of a Hermitian matrix are all real.

[2016, 8M]

2) Let \(A\) be a Hermitian matrix having all distinct Eigen values \(\lambda_{1}\), \(\lambda_{2}\), \(\ldots \lambda_{n}\). If \(X_{1}, X_{2}, \ldots X_{n}\) are corresponding Eigen vectors, then show that the \(n \times n\) matrix \(C\) whose \(k^{t h}\) column consists of the vector \(X_{n}\) is non singular.

[2013, 8M]

3) Let \(H= \begin{bmatrix}{1} & {i} & {2+i} \\ {-i} & {2} & {1-i} \\ {2-i} & {1+i} & {2}\end{bmatrix}\) be a Hermitian matrix. Find a non-singular matrix \(P\) such that \(D=P^{T} H \overline{P}\) is diagonal.

[2012, 20M]

4) Find a Hermitian and skew-Hermitian matrix each whose sum is the matrix \(\begin{bmatrix}{2 i} & {3} & {-1} \\ {1} & {2+3 i} & {2} \\ {-i+1} & {4} & {5 i}\end{bmatrix}\).

[2009, 12M]

Orthogonal and Unitary Matrices

1) Let \(A\) and \(B\) be two orthogonal matrices of same order and \(det\) \(A\) + \(det\) \(B\)=0.Show that \(A+B\) is a singular matrix.

[2019, 15M]

2) Let \(A= \begin{bmatrix}{2} & {2} \\ {1} & {3}\end{bmatrix}\). Find a non-singular matrix \(P\) such that \(P^{-1} A P\) is diagonal matrix.

[2017, 10M]

3) Prove that Eigen values of a unitary matrix have absolute value 1.

[2014, 7M]

4) Find a \(2 \times 2\) real matrix \(A\) which is both orthogonal and skew-symmetric. Can there exist a \(3 \times 3\) real matrix which is both orthogonal and skew-symmetric? Justify your answer.

[2009, 20M]

5) If \(S\) is a skew-Hermitian matrix, then show that \(A=(I+S)(I-S)^{-1}\) is a unitary matrix. Also show that every unitary matrix can be expressed in the above form provided -1 is not an Eigen value of \(A\).

[2005, 15M]

6) If \(\mathrm{H}\) is a Hermitian matrix, then show that \(A=(H+i I)^{-1}(H-i I)\) is a unitary matrix. Also so that every unitary matrix can be expressed in this form, provided 1 is not an Eigen value of \(\mathrm{A}\).

[2003, 15M]

7) If \(A= \begin{bmatrix}{6} & {-2} & {2} \\ {-2} & {3} & {-1} \\ {2} & {-1} & {3}\end{bmatrix}\), then find a diagonal matrix \(D\) and a matrix \(B\) such that \(A=B D B^{\prime}\), where \(B^{\prime}\) denotes the transpose of \(B\).

[2003, 15M]

8) Determine an orthogonal matrix \(\mathrm{P}\) such that \(P^{-1}AP\) is a diagonal matrix, where \(A= \begin{bmatrix}{7} & {4} & {4} \\ {4} & {-8} & {-1} \\ {-4} & {-1} & {-8}\end{bmatrix}\).

[2001, 15M]

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