A tridiagonal matrix is one in which the only non-zero elements are on the principal diagonal, and the two diagonals immediately next to it: one below and the other above.
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The term "eigenvalue" refers to a noun which means each set of values of parameter for which differential equation has a nonzero solution. It can also refers to any number such that given matrix subtracted by the same number and multiply to the identity matrix has a zero determinant.
A sparse matrix is a matrix in which most of the elements are zero.
A zero matrix is a matrix in which all of the entries are zero.