Tagged: complex conjugate
Linear Algebra

05/22/2017

by
Yu
· Published 05/22/2017
· Last modified 11/19/2017

Problem 425
(a) Prove that each complex $n\times n$ matrix $A$ can be written as
\[A=B+iC,\]
where $B$ and $C$ are Hermitian matrices.

(b) Write the complex matrix
\[A=\begin{bmatrix}
i & 6\\
2-i& 1+i
\end{bmatrix}\]
as a sum $A=B+iC$, where $B$ and $C$ are Hermitian matrices.

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Linear Algebra

05/11/2017

by
Yu
· Published 05/11/2017
· Last modified 06/19/2017

Problem 405
Recall that a complex matrix is called Hermitian if $A^*=A$, where $A^*=\bar{A}^{\trans}$.
Prove that every Hermitian matrix $A$ can be written as the sum
\[A=B+iC,\]
where $B$ is a real symmetric matrix and $C$ is a real skew-symmetric matrix.

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Linear Algebra

05/09/2017

by
Yu
· Published 05/09/2017
· Last modified 07/23/2017

Problem 404
Let $A$ be an $n\times n$ real matrix.

Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.

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Linear Algebra

01/22/2017

by
Yu
· Published 01/22/2017
· Last modified 12/02/2017

Problem 269
Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.

(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.

(b) The rank of $A$ is even.

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Linear Algebra

11/27/2016

by
Yu
· Published 11/27/2016
· Last modified 12/02/2017

Problem 202
Show that eigenvalues of a Hermitian matrix $A$ are real numbers.

(The Ohio State University Linear Algebra Exam Problem )
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Linear Algebra

11/21/2016

by
Yu
· Published 11/21/2016
· Last modified 08/11/2017

Problem 191
Let
\[A=\begin{bmatrix}
1 & -1\\
2& 3
\end{bmatrix}.\]

Find the eigenvalues and the eigenvectors of the matrix
\[B=A^4-3A^3+3A^2-2A+8E.\]

(Nagoya University Linear Algebra Exam Problem )
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