A linear function S: U →U is called skew-adjoint if S^*=-S. (a) Prove that a skew-symmetric matr

Step 1: **Prove that a skewu002Dsymmetric matrix is skewu002Dadjoint with respect to the standard dot pr

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A linear function $S: U \rightarrow U$ is called skew-adjoint if $S^=-S$. (a) Prove that a skew-symmetric matrix is skew-adjoint with respect to the standard dot product on $\mathbb{R}^n$. (b) Under what conditions is $S[\mathbf{x}]=A \mathbf{x}$ skew-adjoint with respect to the inner product $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T M \mathbf{y}$ on $\mathbb{R}^n$ ? (c) Let $L: U \rightarrow U$ have an adjoint $L^$. Prove that $L-L^$ is skew-adjoint. (d) Explain why every linear operator $L: U \rightarrow U$ that has an adjoint $L^$ can be written as the sum of a self-adjoint and a skew-adjoint operator.

   A linear function $S: U \rightarrow U$ is called skew-adjoint if $S^*=-S$. (a) Prove that a skew-symmetric matrix is skew-adjoint with respect to the standard dot product on $\mathbb{R}^n$. (b) Under what conditions is $S[\mathbf{x}]=A \mathbf{x}$ skew-adjoint with respect to the inner product $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T M \mathbf{y}$ on $\mathbb{R}^n$ ? (c) Let $L: U \rightarrow U$ have an adjoint $L^*$. Prove that $L-L^*$ is skew-adjoint. (d) Explain why every linear operator $L: U \rightarrow U$ that has an adjoint $L^*$ can be written as the sum of a self-adjoint and a skew-adjoint operator.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 7, Problem 22

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** - A matrix $A$ is skew-symmetric if $A^T = -A$. - To show that $A$ is skew-adjoint, we need to prove that $A^* = -A$, where $A^*$ is the adjoint of $A$ with respect to the standard dot product. - The adjoint $A^*$ of a matrix $A$ with respect to the Show more…

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A linear function $S: U \rightarrow U$ is called skew-adjoint if $S^*=-S$. (a) Prove that a skew-symmetric matrix is skew-adjoint with respect to the standard dot product on $\mathbb{R}^n$. (b) Under what conditions is $S[\mathbf{x}]=A \mathbf{x}$ skew-adjoint with respect to the inner product $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T M \mathbf{y}$ on $\mathbb{R}^n$ ? (c) Let $L: U \rightarrow U$ have an adjoint $L^*$. Prove that $L-L^*$ is skew-adjoint. (d) Explain why every linear operator $L: U \rightarrow U$ that has an adjoint $L^*$ can be written as the sum of a self-adjoint and a skew-adjoint operator.

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