The Inverse Matrix of a Symmetric Matrix whose Diagonal Entries are All Positive

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• Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix 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.   Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
• Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite Suppose $A$ is a positive definite symmetric $n\times n$ matrix. (a) Prove that $A$ is invertible. (b) Prove that $A^{-1}$ is symmetric. (c) Prove that $A^{-1}$ is positive-definite. (MIT, Linear Algebra Exam Problem)   Proof. (a) Prove that $A$ is […]
• Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by \[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where […]
• Find a Matrix that Maps Given Vectors to Given Vectors Suppose that a real matrix $A$ maps each of the following vectors \[\mathbf{x}_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \] into the […]
• Questions About the Trace of a Matrix Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix. (a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]
• Symmetric Matrices and the Product of Two Matrices Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings. (a) The product $AB$ is symmetric if and only if $AB=BA$. (b) If the product $AB$ is a diagonal matrix, then $AB=BA$.   Hint. A matrix $A$ is called symmetric if $A=A^{\trans}$. In […]
• A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$. Hint. Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix. […]
• Quiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]