Introduction to Matrix Proofs

Matrix proofs are an important element of Pure Math and involve establishing the validity of an operation or property using matrices.
Fundamentally, they’re about using the properties of matrices to derive new information or prove given statements.
Understanding matrix proofs is crucial for the study of linear algebra, discrete mathematics, computer graphics, and quantum mechanics.

Types of Matrix Proofs

Direct proofs: Here, the process is straightforward, and involves applying the properties and rules of matrices to confirm the given statement.
Contrapositive proofs: Also known as indirect proofs, these involve proving the negation of the original statement if the original statement is false.
Proof by contradiction: This involves assuming that the statement you’re trying to prove is false and then finding a contradiction that disputes this assumption.

Matrix addition and subtraction: If the matrices are of the same dimensions, then the corresponding entries are added/subtracted.
Scalar multiplication: Every entry in the matrix is multiplied by a constant scalar.
Matrix multiplication: The entry in the i-th row and j-th column of the product is the sum of the products of corresponding entries from the i-th row of the first matrix and j-th column of the second.

For diagonal matrices, the diagonal elements are the eigenvalues of the matrix. This can be proven using characteristic polynomials and determinants.
To prove a matrix A is symmetric, we need to show that A is equal to its transpose, i.e. A = A^T.
To prove that a matrix A is antisymmetric, we must show that A = -A^T.
Proving that a given matrix A is invertible involves demonstrating that it has an inverse A^-1 such that _AA^-1 = _A^-1 A = I, where I is the identity matrix.

Equality of matrices is proven when A = B. Both matrices A and B have to be of the same dimension, i.e., they must have the same number of rows and columns.
Furthermore, the corresponding entries in both matrices must be equal, i.e., if a_ij is an entry in A and b_ij an entry in B, then a_ij = b_ij for all valid i and j.

For proving matrix identities, the best approach depends on the structures of the matrices involved.
Often, it involves applying the rules of matrix arithmetic and sometimes resorting to indicial or positional notation (i.e., working with the individual elements).
Some common matrix identities include AB ≠ BA, (AB)^T = B^T A^T, and (AB)^-1 = B^-1 A^-1 for invertible matrices A and B.
Each of these can be proven using techniques like direct computation, applying definitions, or using properties of matrix operations.