Matrices. Matrices on the TI Creating, Storing, and Displaying Matrices Using Mathematical Functions with Matrices...

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1 13 Matrices Matrices on the TI Creating, Storing, and Displaying Matrices Using Mathematical Functions with Matrices TI -86 M1 M2 M3 M4 M5 F1 F2 F3 F4 F5

2 178 Chapter 13: Matrices Matrices on the TI-86 A matrix is a two-dimensional array, arranged in rows and columns. The matrix elements can be real or complex. You can create, display, and edit matrices on the home screen or in the matrix editor. When you create a matrix, the elements are stored to the matrix name. Creating, Storing, and Displaying Matrices The MATRX (Matrix) Menu - NAMES EDIT MATH OPS CPLX matrix names matrix math complex matrix menu menu menu matrix matrix operations editor menu The TI-86 distinguishes between uppercase and lowercase letters in matrix names. For example, MAT1 and mat1 are two different vector names. The MATRX NAMES Menu - & The MATRX NAMES menu contains all currently stored matrix names in alphanumeric order. To paste a matrix name to the current cursor location, select it from the menu. Creating a Matrix in the Matrix Editor - ' Display the matrix Name= prompt screen. ALPHA-lock is on. The MATRX NAMES menu is displayed. Enter a name from one to eight characters long, starting with a letter. - ' ãmä ãaä ãtä 1 1

3 Chapter 13: Matrices 179 An ellipsis ( ) at either end of matrix rows indicates additional columns. $ or # in the last column indicates additional rows. Display the matrix editor and the matrix editor menu. Accept or change the matrix dimensions (row column) in the top-right corner of the screen, (1 row 255 and 1 column 255); maximum combination is subject to memory availability. The matrix is displayed; all elements are 0. Enter each matrix element value at the element prompt (1,1= for row 1, column 1). You can enter expressions. To move to the next element, press b. To move to the next row, press #. b 10 b 4 b a 4 b 5 b 9 b 6 b 1 b a 3 b 7 b and so on The Matrix Editor Menu - ' matrixname b INSr DELr INSc DELc 4REAL INSr DELr INSc DELc 4REAL Inserts a row at the cursor location; shifts subsequent rows down Deletes row at the cursor location; shifts subsequent rows up Inserts a column at the cursor location; shifts subsequent columns to the right Deletes the column at the cursor location; shifts subsequent columns to the left Converts the displayed complex number matrix to a real number matrix

4 180 Chapter 13: Matrices The close bracket is not necessary when it precedes X. To delete a matrix name from memory, use the MEM DELETE:MATRX screen (Chapter 17). Creating a Matrix on the Home Screen Define the start of the matrix with ã, and then define the start of the first row with another ã. Enter each element for the row, separating them with commas. Define the end of the first row with ä. Define the start of each subsequent row with ã. Enter the row elements, separating each from the next with a comma. Define the end of each row with ä. Then define the end of the matrix with ä. Store the matrix to a matrix name. Either enter a name from one to eight characters long, starting with a letter, or select a name from the MATRX NAMES menu. The matrix is displayed. If newly created, the matrix name becomes a MATRX NAMES menu item P 4 P 6 P a 1 P a 3 P a 5 P a X - n ãmä ãaä ãtä b Creating a Complex Matrix If any matrix element is complex, all elements of the matrix are displayed as complex. For example, when you enter the matrix [[1,2][5,(3,1)]], the TI-86 displays [[(1,0) (2,0)][(5,0) (3,1)]]. To create a complex matrix from two real matrices with the same dimensions, the syntax is: realmatrix+(0,1)imaginarymatrix complexmatrixname realmatrix contains the real part of each element and imaginarymatrix contains the imaginary part of each element.

5 Chapter 13: Matrices 181 To view elements beyond the current screen, use ", #,!, and $. Displaying Matrix Elements, Rows, and Submatrices To display an existing matrix on the home screen, enter the matrix name s individual characters or select it from the MATRX NAMES menu, and then press b. The full value of each element is displayed. Elements with very large values may be expressed exponentially. To display specific elements of matrixname, the syntax is: matrixname(row,column) To display a row of matrixname, the syntax is: matrixname(row) To display a submatrix of matrixname, the syntax is: matrixname(beginrow,begincolumn,endrow,endcolumn) When you execute the expression, the answer is displayed as a matrix. Using a Matrix in an Expression You can enter the matrix directly (for example, 5¹[[2,3][3,5]]). You can use 1 and - n to enter a matrix name s individual characters (for example, MAT1¹3). You can select the matrix name from the MATRX NAMES menu (- &). You can select the matrix name from the VARS MATRX screen (- w / ').

6 182 Chapter 13: Matrices You can use :, 3, and - p to edit matrix elements. You also can overwrite existing characters. Editing Matrices in the Matrix Editor Display the matrix Name= prompt screen. Enter the matrix name. Either select it from the MATRX NAMES menu or enter the characters. Display the matrix editor. Edit or accept the row dimension, and then edit or accept the column dimension. Move the cursor to any element and edit it. Continue moving the cursor to other elements. Save the changes and leave the matrix editor. - ' ãmäãaäãtä 1 1 b 5 3 b 3 b # 45 b " 21 b 2 - ~ b. Editing Matrices on the Home Screen To change a matrix element value, the syntax is: value matrixname(row,column) To change the values of an entire row of elements, the syntax is: [valuea,valueb,...,value n] matrixname(row) To change the values of part of a row, beginning at a specified column, the syntax is: [valuea,valueb,...,value n] matrixname(row,begincolumn) To change the values of a submatrix within matrixname, the syntax is: [[valuea,...,value n]... [valuea,...,value n]] matrixname(beginrow,begincolumn)

7 Chapter 13: Matrices 183 The MATRX MATH Menu - ( NAMES EDIT MATH OPS CPLX det T norm eigvl eigvc 4 rnorm cnorm LU cond det squarematrix matrix T Returns the determinant of squarematrix Returns a transposed matrix; each element s (row,column) coordinates switch norm matrix Returns the Frobenius norm ( G(real 2 +imaginary 2 )) where the sum is over all elements of a real or complex matrix eigvl squarematrix eigvc squarematrix rnorm matrix cnorm Matrix LU(matrix, lmatrixname, umatrixname, pmatrixname) cond squarematrix Returns a list of the normalized eigenvalues of a real or complex squarematrix Returns a matrix containing the eigenvectors for a real or complex squarematrix; each column corresponds to an eigenvalue (row norm) Returns the largest of the sums of the absolute values of the elements (magnitudes of complex elements) in each row of matrix (column norm) Returns the largest of the sums of the absolute values of the elements (magnitudes of complex elements) in each column of matrix Calculates the Crout LU (lower-upper) decomposition of a real or complex matrix; stores the lower triangular matrix to lmatrixname, the upper triangular matrix to umatrixname, and the permutation matrix (which describes the row swaps done during calculation) in pmatrixname Calculates cnorm squarematrix¹cnorm squarematrix M1 ; the closer the product is to 1, the more stable squarematrix can be expected to be in matrix functions

8 184 Chapter 13: Matrices The MATRX OPS (Operations) Menu - ) NAMES EDIT MATH OPS CPLX dim Fill ident ref rref 4 aug rswap radd multr mradd 4 randm Press X to enter the symbol after the close brace. When you use aug(, the number of rows in matrixa must equal the number of rows in matrixb or the number of elements in vector. Elements of matrices created with randm( are integers L9 and 9. dim matrix Returns the dimensions of matrix as a list {rows columns} {rows,columns} dim matrixname Creates a new matrixname of the specified dimensions; each element is 0 {rows,columns} dim matrixname Redimensions matrixname to the specified dimensions Fill(number,matrixName) ident dimension ref matrix rref matrix aug(matrixa,matrixb) aug(matrix,vector) rswap(matrix,rowa,rowb) radd(matrix,rowa,rowb) multr(number,matrix,row) Stores a real or complex number to each matrixname element Returns the square identity matrix of dimension dimension Returns the row-echelon form of matrix Returns the reduced row-echelon form of matrix Concatenates matrixa and matrixb Concatenates matrix and vector Returns a matrix after swapping rowa and rowb of matrix Returns matrix with (rowa+rowb) of matrix stored in rowb Returns matrix with (row¹number) stored in row mradd(number,matrix,rowa,rowb) Returns matrix with ((rowa¹number)+rowb) stored in rowb randm(rows,columns) Creates a matrix of specified dimensions with random elements

9 Chapter 13: Matrices 185 The MATRX CPLX (Complex ) Menu - * NAMES EDIT MATH OPS CPLX conj real imag abs angle To add or subtract two matrices, the dimensions of matrixa must equal the dimensions of matrixb. conj complexmatrix real complexmatrix imag complexmatrix abs matrix angle complexmatrix Returns a matrix in which each element is the complex conjugate of the corresponding element of a complexmatrix Returns a real matrix in which each element is the real portion of the corresponding element of a complexmatrix Returns a real matrix in which each element is the imaginary portion of the corresponding element of a complexmatrix Returns a real matrix in which each element is either the absolute value of the corresponding element of a real matrix or the magnitude (modulus) of the corresponding element of a complex matrix Returns a real matrix in which each element is either 0 if the element of complexmatrix is real or the polar angle if the element of complexmatrix is complex; the polar angles are calculated as tan L1 (imaginary / real) adjusted by +p in the second quadrant and by Lp in the third quadrant Using Mathematical Functions with Matrices matrixa+matrixb matrixanmatrixb Adds each matrixa element to the corresponding matrixb element; returns a matrix of the sums Subtracts each matrixb element from the corresponding matrixa element; returns a matrix of the differences

10 186 Chapter 13: Matrices To multiply two matrices, the column dimension of matrixa must equal the row dimension of matrixb. To enter M1, press - ƒ. Do not use a 1. e^, sin, and cos do not return the exponential, sine, or cosine of each matrix element. To make relational comparisons, matrixa and matrixb must have equal dimensions. == and ƒ are on the TEST menu. round, ipart, fpart, and int are on the MATH NUM menu. matrixa¹matrixb or matrixb¹matrixa matrix¹number or number¹matrix matrix¹vector Mmatrix squarematrix M 1 matrix 2 squarematrix^power e^ squarematrix sin squarematrix cos squarematrix matrixa==matrixb matrixaƒmatrixb Multiplies matrixa and matrixb; returns a square matrix of the products Returns a matrix that is the product of a real or complex number times each element in a real or complex matrix Returns a vector that is the product of each vector element times each matrix element; the matrix column dimension and vector dimension must be equal (negation) Changes the sign of each element in matrix Returns the inverse of squarematrix (not the inverse of each element) Squares a square matrix Raises a squarematrix to the designated power Returns the square matrix exponential of a real squarematrix Returns the square matrix sine of a real squarematrix Returns the square matrix cosine of a real squarematrix Returns 1 if every corresponding element comparison is true; returns 0 if any is false Returns 1 if at least one corresponding element comparison is false round(matrix[,#ofdecimals]) Rounds each matrix element to 12 digits or to specified #of Decimals ipart matrix fpart matrix int matrix Returns the integer part of each element of a real or complex matrix Returns the fractional part of each element of a real or complex matrix Returns the greatest integer of each element of a real or complex matrix