Me, too. I don't know what this means. ##A\cdot 0=0## no matter whether ##A## is invertible or not.ChiralSuperfields said:
The matrix doesn't have an inverse. If we want to prove this by contradiction then we assume it has an inverse. Say the matrix is ##A.## Then ##A## maps the entire space ##\mathbb{R}^2## onto itself: ##A\cdot v= w.## Now, if we set ##v=\begin{pmatrix}x\\ y\end{pmatrix}## then ##A\cdot v=\begin{pmatrix}-x+2y\\-x+2y\end{pmatrix}.## But this means that both coordinates of ##w## are the same and we have no chance to get any other vector with different coordinates. So the image of ##A## is one-dimensional, not two-dimensional, so it cannot be invertible.ChiralSuperfields said:
Eigenvalues are scalar values associated with a square matrix that provide insight into the matrix's properties. Specifically, an eigenvalue is a number λ such that there exists a non-zero vector v (called an eigenvector) for which the matrix A satisfies the equation Av = λv.
To calculate the eigenvalues of a matrix A, you need to solve the characteristic equation det(A - λI) = 0, where det denotes the determinant, I is the identity matrix of the same dimension as A, and λ represents the eigenvalues. The solutions to this equation are the eigenvalues of the matrix.
Eigenvalues have numerous practical applications across various fields. In physics, they can describe the natural frequencies of a system. In computer science, they are used in algorithms for facial recognition and Google's PageRank. In finance, they help in risk assessment and portfolio optimization. Eigenvalues can also indicate stability in systems of differential equations and are used in principal component analysis (PCA) for data dimensionality reduction.
The relationship between eigenvalues and eigenvectors is defined by the equation Av = λv. Here, A is the matrix, λ is an eigenvalue, and v is the corresponding eigenvector. This equation shows that when matrix A acts on eigenvector v, the output is simply the scalar multiplication of v by λ, indicating that v does not change direction under the transformation represented by A.
Yes, a matrix can have complex eigenvalues. This typically occurs when the matrix has complex entries or when the characteristic polynomial has complex roots. Even real matrices can have complex eigenvalues if their characteristic polynomial includes non-real roots. Complex eigenvalues often appear in pairs of conjugates when dealing with real matrices.