9231 · 2.2
Matrices flashcards
Revision flashcards for Cambridge 9231 Matrices (syllabus 2.2). Flip, recall, then mark a real past-paper question.
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What is an eigenvalue of a matrix $\mathbf{A}$?
A scalar $\lambda$ such that for a non-zero vector $\mathbf{x}$, the equation $\mathbf{A}\mathbf{x} = \lambda\mathbf{x}$ holds. It represents a scaling factor.
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What is an eigenvector of a matrix $\mathbf{A}$?
A non-zero vector $\mathbf{x}$ that, when multiplied by $\mathbf{A}$, is simply scaled by a factor $\lambda$ (the eigenvalue), without changing its direction. So, $\mathbf{A}\mathbf{x} = \lambda\mathbf{x}$.
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What is the characteristic equation of a square matrix $\mathbf{A}$?
The polynomial equation in $\lambda$ given by $\det(\mathbf{A} - \lambda\mathbf{I}) = 0$. Its roots are the eigenvalues of $\mathbf{A}$.
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State the Cayley-Hamilton theorem.
Every square matrix satisfies its own characteristic equation. If $p(\lambda) = 0$ is the characteristic equation, then $p(\mathbf{A}) = \mathbf{0}$ (the zero matrix).
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Common Trap: What can an eigenvector NEVER be?
The zero vector, $\mathbf{x} = \mathbf{0}$. The definition requires a non-zero vector, as $\mathbf{A}\mathbf{0} = \lambda\mathbf{0}$ is trivially true for any $\lambda$ and gives no useful information.
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What is the relationship between the eigenvalues and the trace of a matrix?
The sum of the eigenvalues is equal to the trace of the matrix (the sum of the elements on the main diagonal). $\sum \lambda_i = \text{tr}(\mathbf{A})$.
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What is the relationship between the eigenvalues and the determinant of a matrix?
The product of the eigenvalues is equal to the determinant of the matrix. $\prod \lambda_i = \det(\mathbf{A})$.
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How do you find the inverse of a matrix $\mathbf{A}$ using the Cayley-Hamilton theorem?
Substitute $\mathbf{A}$ into its characteristic equation, e.g., $\mathbf{A}^3 + c_2\mathbf{A}^2 + c_1\mathbf{A} + c_0\mathbf{I} = \mathbf{0}$. Multiply by $\mathbf{A}^{-1}$ and rearrange to make $\mathbf{A}^{-1}$ the subject: $\mathbf{A}^{-1} = -\frac{1}{c_0}(\mathbf{A}^2 + c_2\mathbf{A} + c_1\mathbf{I})$.
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What is a key property of the eigenvalues of a real symmetric matrix?
All its eigenvalues are real numbers.
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What is a key property of the eigenvectors of a real symmetric matrix?
Eigenvectors corresponding to distinct eigenvalues are orthogonal (their dot product is zero).
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If $\lambda$ is an eigenvalue of $\mathbf{A}$, what is an eigenvalue of $\mathbf{A}^3$?
$\lambda^3$. In general, if $\lambda$ is an eigenvalue of $\mathbf{A}$, then $\lambda^k$ is an eigenvalue of $\mathbf{A}^k$ for any positive integer $k$.
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What does an eigenvalue of $\lambda=0$ tell you about the matrix $\mathbf{A}$?
It means the determinant of $\mathbf{A}$ is zero ($\det(\mathbf{A}) = \prod \lambda_i = 0$). Therefore, the matrix $\mathbf{A}$ is singular and does not have an inverse.