Tagged: inverse matrix

Sherman-Woodbery Formula for the Inverse Matrix

Problem 250

Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies
\[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\] Define the matrix
\[A=I+\mathbf{u}\mathbf{v}^{\trans}.\]

Prove that $A$ is invertible and the inverse matrix is given by the formula
\[A^{-1}=I-a\mathbf{u}\mathbf{v}^{\trans},\] where
\[a=\frac{1}{1+\mathbf{v}^{\trans}\mathbf{u}}.\] This formula is called the Sherman-Woodberry formula.

 
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Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible.

Problem 216

Let
\[A=\begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}.\] For this problem, you may use the fact that both matrices have the same characteristic polynomial:
\[p_A(\lambda)=p_B(\lambda)=-(\lambda-1)(\lambda+2)^2.\]

(a) Find all eigenvectors of $A$.

(b) Find all eigenvectors of $B$.

(c) Which matrix $A$ or $B$ is diagonalizable?

(d) Diagonalize the matrix stated in (c), i.e., find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$ or $B=PDP^{-1}$.

(Stanford University Linear Algebra Final Exam Problem)
 
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How to Diagonalize a Matrix. Step by Step Explanation.

Problem 211

In this post, we explain how to diagonalize a matrix if it is diagonalizable.

As an example, we solve the following problem.

Diagonalize the matrix
\[A=\begin{bmatrix}
4 & -3 & -3 \\
3 &-2 &-3 \\
-1 & 1 & 2
\end{bmatrix}\] by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(Update 10/15/2017. A new example problem was added.)
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Solving a System of Linear Equations By Using an Inverse Matrix

Problem 65

Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}

(a) Find the coefficient matrix and its inverse matrix.

(b) Using the inverse matrix, solve the system of linear equations.

(The Ohio State University, Linear Algebra Exam)

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Find a Matrix that Maps Given Vectors to Given Vectors

Problem 44

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 vectors
\[\mathbf{y}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \mathbf{y}_2=\begin{bmatrix}
-1 \\
0 \\
3
\end{bmatrix}, \mathbf{y}_3=\begin{bmatrix}
3 \\
1 \\
1
\end{bmatrix},\] respectively.
That is, $A\mathbf{x}_i=\mathbf{y}_i$ for $i=1,2,3$.
Find the matrix $A$.

(Kyoto University Exam)
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