Symmetric and Skew-Symmetric Matrices

Symmetric Matrix

A square matrix $A$ is called symmetric if $A = A^{T}$ , i.e., $a_{j i} = a_{i j}, \forall i, j$ .

Skew-Symmetric Matrix

A square matrix $A$ is called skew-symmetric if $A = - A^{T}$ , i.e., $a_{j i} = - a_{i j}, \forall i, j$ .

Hermitian and Skew-Hermitian Matrices

Hermitian Matrix

A complex square matrix $A$ is called Hermitian if $A = \bar{A^{T}} = A^{H}$ , i.e., $a_{i j} = \bar{a_{j i}}, \forall i, j$ .

Skew-Hermitian Matrix

A complex square matrix $A$ is called skew-Hermitian if $A = - \bar{A^{T}} = - A^{H}$ , i.e., $a_{i j} = - \bar{a_{j i}}, \forall i, j$

Orthogonal and Unitary Matrices

Unitary Matrix

A matrix $A$ is called unitary if $A A^{H} = I$ .

Orthogonal Matrix

A matrix $A$ is called orthogonal if $A A^{T} = I$ .

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 $λ_{1}$ , $λ_{2}$ , $\dots λ_{n}$ . If $X_{1}, X_{2}, \dots X_{n}$ 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 $H = [\begin{matrix} 1 & i & 2 + i \\ - i & 2 & 1 - i \\ 2 - i & 1 + i & 2 \end{matrix}]$ be a Hermitian matrix. Find a non-singular matrix $P$ such that $D = P^{T} H \bar{P}$ is diagonal.

[2012, 20M]

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

[2009, 12M]

Orthogonal and Unitary Matrices

1) Let $A$ and $B$ be two orthogonal matrices of same order and $d e t$ $A$ + $d e t$ $B$ =0.Show that $A + B$ is a singular matrix.

[2019, 15M]

2) Let $A = [\begin{matrix} 2 & 2 \\ 1 & 3 \end{matrix}]$ . 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 $H$ 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 $A$ .

[2003, 15M]

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

[2003, 15M]

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

[2001, 15M]

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