And their friendship and rivalry over almost half a century created an extraordinary visual dialogue to be found at New York's Museum of Modern Art.
Much has been made of the exhibition as a competition, as kind of the heavyweight championship of the 20th century art, says CBS News 60 Minutes Correspondent Morley Safer for CBS News Sunday Morning.
"It's a 15-rounder with no winner," says Kirk Varnedoe, a curator of "Matisse-Picasso." "They're both winners, I think. They both get their hands held up at the end of the game."
They met in 1906 in Paris. Gertrude Stein, the American expatriate poet and patron, whose salon had become the gathering place for the avant-garde of the new century, took Matisse to Picasso's studio in Monmartre. Matisse was 37, an artistic Grandee. Picasso, 25, was a pugnacious upstart.
"Matisse always thought of his art as produced for intellectuals," says Varnedoe. "Picasso is more like a worker, like a mason. He's given himself a workingman's blouse … Instead of the penetrating, intimate gaze that you get in Matisse, there's this kind of thousand-yard stare in Picasso."
They were as different, Matisse said, as the North Pole is to the South Pole. Matisse, a man of northern France, was formal and austere. Picasso, born in southern Spain, was spirited and impulsive.
Matisse was the figurative painter but obsessed with explosive color. Picasso was the conceptual painter who emphasized line and form.
However it was Matisse's shocking "Blue Nude: Memory of Biskra," of 1907, whose distortion of the female figure, John Elderfield says, challenged Picasso.
"It looks ugly and maybe deformed now, and it's hard to imagine what it looked like in 1907," he explains. "So Picasso, who was already a more aggressive artist, to see this … I think pushed him along."
And Picasso produced "ugly" and startling figures. The dramatic pairing of Picasso's breakthrough edgy masterpiece "Les Demoiselles D'Avignon" of 1907, and Matisse's enigmatic "Bathers with a Turtle" of 1908, are two of the most extraordinary works in the show.
"'Les Demoiselles D'Avignon,' which is a scene of prostitutes in a bordello, was conceived by Picasso, I think, as a kind of anti-Matisse," says Elderfield. "Matisse painted outdoor scenes of lyrical beauty … Matisse makes sexuality lyrical and all about love."
It is said that when Matisse saw the "Les Demoiselles D'Avignon" he was appalled because it seemed in violation of all of the canons of beauty, and Matisse thought it was a farce. Sham or not, Matisse had to deal with it.
By 1908, cubism intrudes into the dialogue. Matisse's "Portrait of Madame Matisse" is a quiet homage compared to Picasso's astonishing "Woman in Yellow" or his brutal "Portrait of a Young Girl," where the figure is reduced to a kind of insane geometry.
"This was Picasso's invention," says Elderfield. "He had made a kind of decomposed painting like 'The Guitar Player,' of the kind of painting no one had ever seen before. I mean, it's difficult to decipher. The forms are pulled apart. Matisse had great difficulty with this kind of painting and didn't really know how to respond to it."
He wasn't alone. A lot of the public did were just as perplexed — trying to see where the guitar is in "the guitar player."
"I think that cubism pushes Matisse to extend his notion of himself," says Elderfield. "In this period from 1914-1917, he really mixed some pictures which people do think to be profoundly anti-Matissean, made by Matisse."
Matisse's pictures, such as the "Goldfish and Palette," painted in 1914, made some believe it inspired Picasso's "Harlequin" of 1915.
"When you look at them side by side, you can see that Matisse's use of black, this stunning big slab of black down the middle of the goldfish, and the austerity of the picture does have its echo in Picasso's 'Harlequin.'"
But Matisse would never say Picasso influenced his works. But Matisse's painting of a tectonic structure of big, black bars shows how a Picasso miniature style can be monumental and enormous.
Elderfield says the two artists poached onto each other's territory to make something different that the other hadn't realized.
"So they're constantly demonstrating to the other, 'I can do what you're doing, better,'" says Elderfield.
As in Picasso's "The Three Dancers" and Matisse's "Nasturtiums with 'Dance' II."
"The Picasso 'Dancers' takes a theme that belongs to Matisse," explains Elderfield. "There's a kind of Dionysian frenzy, particularly the dancer on the left who's truly wild and disheveled. So that it has a great energy."
But he says the savage, angular, passionate and tragic style of the painting replaces the lyricism and beauty of Matisse's work.
Could one artist be better without the other?
"Neither one of them would have been what they were without the other," says Elderfield. "I think that the competition and the stimulation of seeing the other's work made each of them achieve things they never would have achieved on their own."
By 1918, they'd drifted apart. Picasso was painting weighty women and lascivious satyrs, probably self-portraits. Matisse, worked on luxurious portraits of women of odalisques, exotic harem-girls. But even then, they kept an eye on each other.
In the 1920s, Picasso began pushing the edge in sculpture after he saw Matisse's early works. Picasso's cubism painting influenced Matisse's sculptures. After seeing Matisse's sculptures, Picasso sculpted work similar to his painting style.
In 1932, everything changes. A new muse enters Picasso's life. Her name was Marie-Therese Walter.
"He really is erotically stimulated, and he wants to go back and celebrate the ripeness and fullness of female beauty," says Elderfield. "So he turns to Matisse. There's a kind of dialogue where Picasso takes over Matisse's subject and teaches Matisse a thing or two about how to remake it."
Matisse painted "Music" in 1939, just as World War II began, followed by "Asia" in 1946 – brighter and more colorful compositions next to Picasso's "Serenade" of 1942.
Matisse's work during times of war had shut the reality out and focuses on the beauty of the world. Picasso, in comparison, acknowledged the threat and ugliness around him.
In 1948, Picasso joined Matisse in the South of France. Matisse's health was in decline. He could barely paint, so he turned to scissors and paper — proving that he still had the stuff of genius.
"The scissors made him rediscover a new way of drawing," says Elderfield. "It was like a combination of sculpting and drawing all at the same time, to cut the form with the scissors."
Elderfield believes Picasso learned from watching Matisse work with cut paper sculptures.
Henri Matisse died in 1954 at the age of 84. And in 1973, Picasso died age of 91.
As Kirk Varnedoe said, the heavyweight championship of the 20th century was fought to a draw. Before they died, these two men, both suffused with immeasurable talent and ego, paid tribute to each other. Picasso said, "All things considered, there is only Matisse." Matisse from his austere ivory tower, proclaimed: "Only one person has the right to criticize me. It is Picasso."