Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix
Problem 210
Let $A$ be an $n\times n$ matrix with real number entries.
Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.
Add to solve later Tagged: transpose
Add to solve later Subspaces of Symmetric, Skew-Symmetric Matrices
Problem 143
Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.
(a) The set $S$ consisting of all $n\times n$ symmetric matrices.
(b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices.
(c) The set $U$ consisting of all $n\times n$ nonsingular matrices.
Add to solve later Rank and Nullity of a Matrix, Nullity of Transpose
Problem 140
Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.
(a) $\calN(A)=\calN(A^{\trans}A)$.
(b) $\rk(A)=\rk(A^{\trans}A)$.
Add to solve later Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Problem 136
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
Add to solve later 10 True or False Problems about Basic Matrix Operations
Problem 104
Test your understanding of basic properties of matrix operations.
There are 10 True or False Quiz Problems.
These 10 problems are very common and essential.
So make sure to understand these and don’t lose a point if any of these is your exam problems.
(These are actual exam problems at the Ohio State University.)
You can take the quiz as many times as you like.
The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.
Add to solve later Trace of the Inverse Matrix of a Finite Order Matrix
Problem 46
Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix.
Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that $\tr((A^{-1})^{\trans})=\overline{\tr(A)}$.
Add to solve later Find a Basis of the Subspace of All Vectors that are Perpendicular to the Columns of the Matrix
Problem 40
Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix
\[A=\begin{bmatrix}
11 & 12 & 13 & 14 \\
21 &22 & 23 & 24 \\
31 & 32 & 33 & 34 \\
41 & 42 & 43 & 44
\end{bmatrix}.\]
(Harvard University Exam)
Add to solve later If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible
Problem 38
Let $A$ be an $m \times n$ real matrix.
Then the kernel of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$.
The kernel is also called the null space of $A$.
Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is invertible.
(Stanford University Linear Algebra Exam)
Add to solve later Transpose of a Matrix and Eigenvalues and Related Questions
Problem 12
Let $A$ be an $n \times n$ real matrix. Prove the followings.
(a) The matrix $AA^{\trans}$ is a symmetric matrix.
(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.
(c) The matrix $AA^{\trans}$ is non-negative definite.
(An $n\times n$ matrix $B$ is called non-negative definite if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)
(d) All the eigenvalues of $AA^{\trans}$ is non-negative.
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