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True or False: Eigenvalues of a Real Matrix Are Real Numbers
Answer the following questions regarding eigenvalues of a real matrix.
(a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.
(b) Find the eigenvalues of the matrix
\[B=\begin{bmatrix}
-2 & -1\\
5& 2
\end{bmatrix}.\]
(The Ohio State University, Linear Algebra Exam)
Add to solve later
Hint.
Consider a $2\times 2$ matrix.
Then the eigenvalues are solutions of a quadratic polynomial.
Does a quadratic polynomial always have real solutions?
False. In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.
The characteristic polynomial for $B$ is
\[ \det(B-tI)=\begin{bmatrix}
-2-t & -1\\
5& 2-t
\end{bmatrix}=t^2+1.\]
The eigenvalues are the solutions of the characteristic polynomial. Thus solving $t^2+1=0$, we obtain eigenvalues $\pm i$, where $i=\sqrt{-1}$.
Thus the eigenvalue of a real matrix $B$ is pure imaginary numbers $\pm i$.
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Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
How to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression \[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right. \] where $a,b,c, d$ […]
Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix Let $A$ be an $n\times n$ matrix with the characteristic polynomial \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable. (a) Find the size of the matrix $A$. (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue […]
Given the Characteristic Polynomial, Find the Rank of the Matrix Let $A$ be a square matrix and its characteristic polynomial is given by \[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\] Find the rank of $A$. (The Ohio State University, Linear Algebra Final Exam Problem) Solution. Note that the degree of the characteristic polynomial […]
Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix? (The Ohio State University, Linear Algebra Final Exam […]
Solving a System of Linear Equations By Using an Inverse Matrix 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 […]
Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations Consider the matrix \[A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.\] (a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why. (b) Are the vectors \[ […]
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Introduction to Matrices
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Gaussian-Jordan Elimination
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Linear Combination and Linear Independence
Nonsingular Matrices
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Bases and Dimension of Subspaces in $\R^n$
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Linearly Independency of General Vectors
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Linear Transformation from $\R^n$ to $\R^m$
Linear Transformation Between Vector Spaces
Orthogonal Bases
Determinants of Matrices
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Introduction to Eigenvalues and Eigenvectors
Eigenvectors and Eigenspaces
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Dot Products and Length of Vectors
Eigenvalues and Eigenvectors of Linear Transformations
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Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations
Consider the matrix \[A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.\]...