Rank, row reduction, trace and eigenvalues of a matrix in Scientific calculator plus 991 for Android

These are the functions a linear-algebra course reaches after the determinant: how many rows of a matrix are independent, the simplest form row operations can bring it to, the sum of its diagonal, and the numbers and directions that describe what it does. Each has a key on the 2nd keyboard, and MATRIX Mode lists them in the MATH section of its menu, where they take a stored matrix by name.

Rank

The rank counts how many rows are genuinely independent of each other. On the 2nd keyboard, the key is marked Rank but inserts MatrixRank(, which is the name to look up in the app's help. The manual's R is [[1, 2, 3], [4, 5, 6]], and both its rows are independent:

Here the second row is just twice the first, so only one row is independent:

In MATRIX Mode, MatrixRank(matrix) is in the MATH section and takes a stored matrix. The manual's MatC, [[1, 2, 3], [4, 5, 6], [7, 8, 10]], has no redundant row:

Row reduction: solve a system

RowReduce(matrix) reduces a matrix to its simplest row form. It is the quickest way to solve a system of linear equations by row operations: write the coefficients and the constants as one matrix, reduce it, and read the answers from the last column.

The left part of the answer is the identity, so each row now says one unknown on its own: 1x + 0y = 1 and 0x + 1y = 2. The last column reads 1 and 2, so x = 1 and y = 2.

When the values are all you need, the simultaneous-equation mode takes the same rows as a grid of coefficients, for two to four unknowns, and gives each value in turn. RowReduce shows the reduced matrix itself, which is what a question on row operations asks to see.

Trace and diagonal

The trace is the sum of the elements down the main diagonal. On the 2nd keyboard it is SHIFT Rank Trace, which inserts Tr(. With the manual's P, [[1, 2], [3, 4]], that is 1 + 4:

Unlike the determinant, it does not need a square matrix; it simply adds whatever lies on the diagonal. For R, [[1, 2, 3], [4, 5, 6]], that is 1 + 5:

Diagonal(matrix) lists the elements Tr adds up. It has no key: choose it from the linear algebra menu.

Eigenvalues and eigenvectors

The eigenvalues of a square matrix are the numbers by which it stretches its own special directions, and the eigenvectors are those directions. One key on the 2nd keyboard gives both: EiValues inserts Eigenvalues(, and SHIFT EiValues EiVectors inserts Eigenvectors(.

For the eigenvectors, press SHIFT EiValues EiVectors in the first step instead. Each row of the answer is the direction belonging to one eigenvalue, in the same order as the eigenvalues came out:

So (1, 1) belongs to the eigenvalue 3, and (−1, 1) to the eigenvalue 1. Both functions need a square matrix.

In MATRIX Mode, Eigenvalues(matrix) and Eigenvectors(matrix) are in the MATH section too, and take a stored matrix by name.

The linear algebra menu

The linear algebra key on the 2nd keyboard opens a list of about sixty linear-algebra functions, everything above and much more; type in the search box at the top to filter it. For longer work it carries, among others:

• CharacteristicPolynomial(matrix, var), the characteristic polynomial, whose roots are the eigenvalues.
• LinearSolve(matrix, right), which solves a system directly, and LeastSquares(matrix, right), the best fit for an over-determined one.
• NullSpace(matrix) and PseudoInverse(matrix).
• LUDecomposition(matrix), QRDecomposition(matrix) and SingularValueDecomposition(matrix).
• DiagonalMatrix(list), which builds a matrix from its diagonal, with Adjugate(matrix), Minors(matrix), MatrixPower(matrix, n) and MatrixExp(matrix).

MATRIX Mode's MATH section lists CharacteristicPolynomial, Diagonal, DiagonalMatrix, LeastSquares, LinearSolve, LUDecomposition, SingularValueDecomposition, NullSpace and PseudoInverse as well.

To fit a line or a curve to paired data rather than solve an exact system, use regression and curve fitting in STAT mode instead.

At a glance