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How can we compute the distance between two matrices if the matrix dimensions do not match?
You just take the m rows of P, place them one after the other to form a row vector of length mn; do the same with the m rows of Q; then use the ordinary dot product in R^{mn}. Another way to say this: Multiply each entry of P by the corresponding entry of Q and add the mn products just created.
Are all triangular matrices diagonalizable?
For these two cases the diagonalizability of upper triangle matrix A can be recognized “by inspection”: If all diagonal entries are distinct, A is diagonalizable. If all diagonal entries are equal, A is diagonalizable only if A itself is diagonal, as shown in Diagonalizable properties of triangular matrix.
How to calculate the similarity of a matrix?
Understand the relationship between the eigenvalues, eigenvectors, and characteristic polynomials of similar matrices. Recipe: compute Ax in terms of B , C for A = CBC − 1 . Picture: the geometry of similar matrices. Vocabulary word: similarity. Some matrices are easy to understand. For instance, a diagonal matrix
Which is the only matrix similar to i n?
Any invertible matrix is row equivalent to I n , but I n is the only matrix similar to I n . For instance, are row equivalent but not similar. As suggested by its name, similarity is what is called an equivalence relation. This means that it satisfies the following properties.
When are the determinants of two matrices are the same?
If Two Matrices are Similar, then their Determinants are the Same Prove that if A and B are similar matrices, then their determinants are the same. Proof. Suppose that A and B are similar. Then there exists a nonsingular matrix S such that S − 1AS = B by definition.
Can a matrix similar to a be diagonalizable?
A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let A, B be matrices. Show that if A is diagonalizable and if B is similar to A, then B is diagonalizable. Definitions/Hint.